Resonator State Preparation with Rydberg Atom Beams

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HYBRID QUANTUM SYSTEMS: RESONATOR STATE PREPARATION WITH RYDBERG ATOM BEAMS

Robert Jirschik

Supervisor: Prof. Stephen Hogan

Co-supervisor: Prof. Sougato Bose

Department of Physics and Astronomy

University College London

20.05.2019

A dissertation submitted to University College London for the degree of Doctor of Philosophy Declaration

I, Robert Jirschik, conﬁrm that the work presented in this thesis is my own. Where information has been derived from other sources, I conﬁrm that this has been indicated in the thesis.

Date: ...... Signature: ......

2 Abstract

In this thesis theoretical studies of hybrid quantum systems composed of solid-state superconducting microwave resonators, mechanical oscillators, and gas-phase Rydberg atoms are described. The thesis begins with an overview of the current state of the ﬁeld of hybrid quantum information processing. An introduction to the physics of Rydberg atoms and the preparation processes of circular Rydberg states follows, including a review of the associated literature. Three main new research results are then presented. These include (1) numerical studies of static electric dipole interactions in strongly polarized gases of Rydberg atoms. The understanding and characterization of these interactions is essential to maximize the quantum-state preparation procedures required for the hybrid systems studied in the thesis; (2) analytical and numerical studies of quantum-state preparation and cooling in coplanar superconducting microwave resonators using beams of atoms in circular Rydberg states; and (3) studies of coupled Rydberg-atom—mechanical-oscillator systems. The results presented in each of these areas are of interest for hybrid quantum information processing and quantum computation, and optical-to- microwave photon conversion for quantum communication.

3 Impact Statement

The research reported in this thesis is of interest in several research areas. Approaches to the preparation of quantum states of resonator systems in a controlled manner as presented here have applications in several quantum technologies. For example, the selective preparation of the vacuum state of the resonator can be used to increase the ﬁdelity of quantum state transfer protocols and thus be applied in the context of quantum communication. The study of electrostatic dipole-dipole interactions within ensembles of Rydberg atoms described in this thesis further the understanding of many- body physics in strongly polarized gases. The combination of this study and the resonator state preparation results could lead to new quantum measurement techniques to characterize atomic ensembles, as well as the realization of quantum non-demolition measurements of gas-phase atoms and molecules, including exotic species such as antihydrogen and positronium. In the long term, the results presented here constitute a stepping stone in the development of a hybrid quantum information processing architecture with transmission-line microwave resonators controlled and manipulated using Rydberg atoms. This could lead to the development of new, robust quantum simulators for studying complex condensed matter problems that cannot be solved using classical computation technology.

4 Acknowledgements

There are too many people to thank for all the support I received during my time as a PhD student. I shall give it a try here. First and foremost I would like to thank my supervisor, Prof. Stephen Hogan, for giving me the opportunity to work with him. His passion for science, his dedication, interest and enthusiasm with which he approaches his work all have been a great motivation for me. I could not have ﬁnished this thesis without his support, ideas, suggestions and discussions. I would also like to thank my Co-supervisor, Prof. Sougato Bose, for his cre- ative input and help. Another thank you to my Diplomarbeit-supervisor Prof. Michael Hartmann. My overwhelmingly positive experience during my research year changed my perspective on physics, and encouraged me to pursue a PhD. A special thanks to my parents, Kathrin and Dirk, who have been encour- aging me to follow my passions without a worry. They are amazing role models. I cannot thank them enough for their advice and moral support. I would also like to thank a few PhD students and Postdocs who have passed through our oﬃce over the years, or with whom I had the plea- sure of discussing our work: Hoda Hossein-Nejad, Patrick Lancuba, Ed- ward O’Reilly, David Holdaway, Richard Stones, Valentina Zhelyazkova,

5 Valentina Notararigo, Stefan Siwiak-Jaszek, Adam Deller, James Palmer and Alex Morgan. My apologies to everyone else I have interacted with at UCL and did not mention here, your contributions are all appreciated. And ﬁnally, a big thank you to all my close friends. It seems unfair to mention only some of them by name, but a special thank you to my clos- est friend Felix for his unrivaled reliability, and the two people with the biggest inﬂuence on me during my time in London, Renaud and Olivier. Without their distractions this endeavour would not have been even half as fun.

6 Contents

1 Introduction 22

2 Rydberg Atoms 29 2.1 Hydrogen atoms in Rydberg States ...... 29 2.2 Helium atoms in Rydberg states ...... 33 2.3 Atoms in Circular Rydberg States ...... 33 2.3.1 Helium Atoms in Circular Rydberg States ...... 40

3 Electrostatic Dipole Interactions in Polarized Rydberg Gases 44 3.1 Experiment ...... 47 3.2 Results ...... 51 3.3 Simulations ...... 53 3.4 Conclusions ...... 58

4 Superconducting Transmission Line Resonator 60 4.1 Quantization of a 1D Transmission Line Resonator ...... 61 4.2 Dissipation from a 1D Transmission Line Resonator . . . . . 65

5 Cooling a transmission-line resonator with a beam of Rydberg atoms 72

7 5.1 The atomic beam as an engineered quantum reservoir ...... 73 5.2 Resonator De-Excitation ...... 81 5.2.1 Steady State Solution ...... 85 5.2.2 Time-Resolved Solution ...... 94

6 Atomic Beam Interaction: Numerical Approach 100 6.1 Modeling the Beam of Atoms ...... 101 6.2 Stochastic Dynamics ...... 106 6.2.1 The stochastic process as a series of time steps . . . . 107 6.2.2 The dynamics for a constant number of atoms . . . . 108 6.2.3 A beam of atoms interacting with the resonator . . . 109 6.2.4 Ensemble average of the trajectories ...... 111

7 A Microwave Resonator-Nanomechanical Oscillator–System driven by an Atomic Beam 117 7.1 Derivation of the Hamiltonian ...... 118 7.2 Microwave Resonator-Mechanical Oscillator–Subsystem . . 122 7.3 Microwave Resonator-Mechanical Oscillator-Atom–System . 125 7.4 The Atomic Beam as an engineered environment ...... 130 7.4.1 Steady State of the system interacting with an atomic beam ...... 135 7.4.2 Considering a Thermal Environment ...... 138

8 Conclusions 143

A Probability of N Atoms interacting with the Resonator 164

8 List of Figures

1.1 Overview of hybrid quantum systems. Each individual system is positioned according to its excitation frequency and coherence time, with possible couplings between them indicated by arrows. From [65]...... 24

2.1 The helium atom in a high n Rydberg state. The outer electron "sees" the ion core and inner electron as one central ﬁeld with charge Z = +1. The Coloumb potential associated with the He2+ nucleus is screened by the inner electron...... 34

2.2 The effect of the quantum defects δn,` on the binding energy of the atomic states for non-hydrogenic atoms compared to the hydrogen atom. From [53]...... 35 2.3 Adiabatically switching off the electric field, F~, leads to a rotation of the vector operator ~λ. This is equivalent to the evolution of the upper state’s parabolic ellipse, with angular

momentum m = 0, to a circular trajectory with m = Lz = n 1. 39 − 2.4 The energy level diagrams for steps 2-4 in the preparation process of circular Rydberg states for n = 70 in Helium as described in the text. From [80]...... 41

9 3.1 (a) Schematic diagram of the photoexcitation and detection region of the experimental apparatus. The position of laser photoexcitation and the approximate spatial extent of the Rydberg atom ensemble in the y dimension during microwave interrogation are indicated by the red circle and dashed rectangle, respectively. (b) Sequence of events in each experimental cycle, including laser photoexcitation, Rydberg state polarization, microwave interrogation, and state-selective electric ﬁeld ionization (not to scale). From [128]...... 48

3.2 Stark structure of the triplet m` = 0 Rydberg states of He with values of n ranging from 70 to 72. Experiments were performed following laser photoexcitation of the 70s state in zero electric ﬁeld and the subsequent polarization of the

1 excited ensemble of atoms in ﬁelds of (i) 18 mV cm− , (ii)

1 1 358 mV cm− , and (iii) 714 mV cm− as indicated. From [128]. 50

10 3.3 Experimentally recorded (continuous curves) and calculated (dashed curves) spectra of transitions between the Stark

3 3 states that evolve adiabatically to the 1s70s S1 and 1s72s S1 levels in zero electric ﬁeld. The experiments were performed at low (blue curves) and high (red curves) Rydberg atom number densities, and spectra were recorded in polarizing

1 1 electric ﬁelds of (i) 18 mV cm− , (ii) 358 mV cm− , and (iii)

1 714 mV cm− , for which µ70 = 380 D, 7250 D, and 12250 D

(µ72 = 470 D, 8650 D, and 14050 D), respectively. The calculated spectra are presented (a) without, and (b) with effects of the local polarization of the medium on the macroscopic dielectric properties accounted for. The relative microwave frequencies on the horizontal axes are displayed with respect to the transition frequencies [(i) 37.2413 GHz, (ii) 37.1068 GHz, and (iii) 36.8094 GHz] recorded at low Ry- dberg atom number density. From [128]...... 52 3.4 Experimentally recorded (continuous curves) and calculated (dashed curves) dependence of the n = 70 n = 72 spectra → of the most strongly polarized ensemble of atoms, for which

µ70 = 12 250 D and µ72 = 14050 D, on the Rydberg atom density. From [128]...... 57

11 4.1 Simple LC-circuit electrical oscillator: (a) In order to derive a Lagrangian where the single degree of freedom is the charge, q, the usual sign convention for the flow of the current, I, from q to q can be used. (b) The sign convention in the − text, where the positive and negative signs of the charge, Q, have been flipped with respect to the current, leads to the magnetic flow as the single degree of freedom in the Lagrangian describing the LC-circuit. From [41]...... 61 4.2 The resonator-waveguide system and its lumped circuit ana- logue. The resonator and waveguide can be considered as a sum of harmonic LC-circuits, where coupling capactiances at each end impose boundary conditions...... 66 4.3 Comparison of the here calculated Q-factor and the measurement by Goeppl et al. Both curves show qualitatively similar behavior, whereas the precise values differ because the details of the experimental parameters of the waveguide could only be estimated from the published information. . . 70

12 5.1 The population transfer rates An, Bn and Cn among three consecutive Fock states in the resonator. It can be seen that

Bn corresponds to a de-excitation of the resonator from the

and Cn 1 populating the n Fock state, the rate of change of − | i the population of that state n can be calculated according | i to Eq. (5.28)...... 83 5.2 Distribution of the photon occupation number for a fundamental frequency of ν = 18.78 GHz. The red dots mark the occupation probability of the Fock state n of the resonator | i in the non-equilibrium steady state. The Planck distribu-

tion for an effective temperature Teff = 360 mK is plotted (blue line), showing the qualitative fit with the steady state photon distribution...... 86

13 5.3 Depletion of the fundamental mode of the resonator. The thinner red bars represent the occupation probability distribution of the Fock states n at T = 4 K. This results in | i an average photon occupation number of n = 3.96. The h ith thicker blue bars represent the photon number distribution after the interaction with an atomic beam with a fixed atom- resonator interaction time τ¯, leading to a thermal photon number distribution with an average number of photons n = 0.122...... 87 h iss 5.4 The dependence of the 0 Fock state fidelity on the detun- | i ing of the atomic transition frequency and the resonator resonance frequency of the selected fundamental mode. . . . 89 5.5 The dependence of the resonator ground state fidelity on the atom-resonator coupling strength. A coherent beam (solid blue line) reaches local minima for specific values of the interaction strength, one being at g = 62.82 MHz. The minima represent trapping states. The incoherent atomic beam (dashed red line) does not exhibit such behavior, with a roughly constant fidelity of 0.85 for values of g 80 F0 ≈ ≥ MHz...... 90 5.6 Photon number distribution corresponding to the annomalous trapping state that arrises for an atom-resonator coupling strength of g = 62.82 MHz...... 92

14 5.7 The dependence of the steady state average photon number , n , on the resonator quality factor, Q. The product h iss rτ 1 ensures the approximation of at most one atom interacting with the resonator is fulfilled...... 93 5.8 Dependence of the average photon number n of the nonequi- h i librium steady state of the resonator on the rate of atoms crossing the resonator with different average interaction times τ¯ as indicated. The continuous curves represent the values for r that fulfill the approximation rτ 1, where the probability of having two or more atoms interact with the resonator at the same is less than 0.25%. The dashed curves represent the region in which this approximation breaks down because the probability that two or more atoms interact with the resonator at the same time is not negligible anymore...... 95 5.9 Time-dependent evolution of the average photon number n of the resonator. The initial state of the resonator with a h i fundamental frequency of ν = 18.78 MHz is assumed to be a

thermal state at Tenv = 4 K. The atoms on resonance, ∆ = 0,

6 1 cross the resonator at a rate r = 10 s− and couple to it for a constant interaction time τ¯ = 50 ns with a coupling strength g = 26.66 MHz...... 98 5.10 Time resolved reheating of the resonator...... 99

15 6.1 A schematic diagram of the experimental setup, allowing for more than one atom to interact with the microwave resonator at any given time t...... 101 6.2 The ﬂow chart of the numerical approach. Within each time step ∆t the number of atoms leaving the microwave resonator is calculated (orange arrows) after having interacted

over a time tint. This depends on how many atoms remain within the resonator, and how many new atoms enter the microwave resonator (black arrows), based on their arrival

probability Pn(∆t). The black arrows from subsequent orange atom number boxes, have been omitted for clarity of the sketch...... 103 6.3 The probability distribution representing the number of atoms interacting with a resonator over a time T = 2 µs, with an average interaction time of τ¯ = 10 ns and a rate of r = 40 atoms/µs. The simulation was carried out for 500 trajectories...... 106

16 6.4 Schematic diagram of the average photon occupation number of the microwave resonator with atoms being deterministically monitored for a single realization. The moments atoms enter the resonator are marked by grey, dashed gridlines. Three atoms enter at a time t = 0, the state of one of them being displayed by the dashed blue curve. Two atoms leave and one enters at t = 500. At a time t = 1000, the atom 1 leaves the resonator, and a diﬀerent atom 2 enters, whose dynamics are displayed by green dashed curves, and two additional atoms are introduced. At t = 1500 the same atom remains and its state is displayed, while 2 new atoms enter, one of which is being displayed by the red dashed curve. . . 110 6.5 Comparison of the analytical and numerical method of treating the coupled atom-resonator system in a parameter regime in which both approaches are valid. The rate of Rydberg

6 1 atoms crossing the resonator is r = 10 s− , the constant interaction time is τ = 50 ns, and the interaction strength is g/(2π) = 4.25 MHz...... 113 6.6 The eﬀect of a dense beam of atoms prepared in the lower of two circular Rydberg states on the thermal photon occupation number in the resonator. The rate of Rydberg atoms crossing the resonator is r = 15 MHz, the constant interaction time is τ = 50 ns, and the interaction strength is g/(2π) = 4.25 MHz...... 114

17 6.7 Two calculated trajectories for individual atoms compared to the average of 100 simulated trajectories, displaying the quantum jumps in the photon field induced by an atom entering the resonator and being excited...... 115 6.8 Comparison of the photon probability distribution of the method a) studied by Sarkany et al. [99] and b) the method investigated in this chapter with an atomic rate of r = 145 · 6 1 10 s− and an average interaction time of τ¯ = 10 ns. Both concepts lead to a sharp peak around n = 0 and a significantly lower average effective photon number in the steady state...... 115 6.9 Time-dependence of the average resonator photon number. An average steady state photon number of n & 0.1 can be h iss reached within a time of t 1 µs, utilizing approximately ≈ 150 Rydberg atoms to effectively cool the resonator...... 116

7.1 (1) The dependence of the eigenenergies of both polariton- modes is displayed when the detuning between the mi-

crowave resonator and the laser drive ∆˜ 0 is decreased from

2 ωm to 0 ωm by varying the laser drive frequency. (2a) and (2b) show the relative amplitudes of the photonic and phononic excitation...... 124

18 7.2 The ﬁrst 3 eigenenergies of the Hamiltonian Hi depending on the detuning between the microwave resonator detuning

and the mechanical oscillator ∆ = ∆˜ ωm. The detuning 0 − between the microwave resonator and the laser drive ∆˜ 0 is

decreased from 2 ωm to 0 ωm...... 127

7.3 The quasiparticle-composition of the eigenenergies (a) ωx,

(b) ωy and (c) ωz in Equ. (7.29) of the interaction Hamiltonian,

Hi, that are plotted in Fig. 7.2...... 129 7.4 The eﬀective cooling process for the polariton excitations

α†α + β†β . Note that the x-axis is inversed in order to h iss h iss illustrate the eﬀect for an increase in the rate of atoms prepared in the lower circular Rydberg state. The inset shows the same calculation with a linear y-axis...... 137 7.5 The dependence of the two polariton branches on the rate

of the atomic beam,r, for Nm = 40. Because γr/γm = 5, the average polariton numbers of the initial, thermal state are

α†α 4.74 (mainly photonic mode) and β†β 9.17 h ith ≈ h ith ≈ (mainly phononic mode)...... 142

A.1 The dependence of the probability of more than ﬁve atoms interacting with the resonator at the same time on the product of r τ. The dashed lines indicate the maximum value of · the product r τ for which the error is 0.5%...... 168 · ≤

19 List of Tables

1.1 Important properties for diﬀerent systems that can be used as qubits [49, 57, 80, 81, 83, 121]...... 23 1.2 Important properties for diﬀerent types of resonators [57, 42, 81, 63, 25]...... 25

2.1 Comparison between some of the properties of low- and high-angular-momentum Rydberg states...... 37

4.1 The values used for the theoretical calculation of the Q-factor as published by Goeppl et al. [42]...... 70

20 "He helps you to understand He does everything he can Doctor Robert" - The Beatles

21 Chapter 1

Introduction

In recent years the development of quantum technologies has made considerable progress. The ability to control the quantum dynamics in experiments with atoms and trapped ions has been greatly enhanced and culminated in the award of the Nobel Prize in Physics to Serge Haroche and David J. Wineland in 2012 for their work in cavity quantum electrodynamics [47, 119]. In addition to this discipline within quantum optics, the development of superconducting quantum electrical circuits [124, 31, 113] in the last two decades and the proposal to, i.e., employ such systems as quantum simulators for light-matter interactions [12], has further been aid- ing the progress of the ﬁeld of quantum information processing because of their scalability and control. Other systems have also been emerg- ing as promising candidates as quantum bits for quantum information processing due to their long coherence times, such as ensembles of electron spins [120], nitrogen-vacancies (NV) in diamonds [7], and quantum dots in semiconductors [90]. Additionally, the use of micro- and nanomechanical oscillators, i.e., in the form of microtoroidal resonators, carbon

22 nanotubes or membranes, and the ability to reach the quantized regime of their mechanical motion have been oﬀering new possibilities for tests of fundamental quantum physics [15, 74, 25], and for quantum information processing [96]. All of these individual quantum systems, that contain qubits and resonators, oﬀer distinct advantages and disadvantages, as outlined in Table 1.1 and Table 1.2.

n = 70 Optical Electronic SC Transmon Rydberg atom Ion spin ensembles qubit Energy 20 GHz 105 GHz 1 10 GHz 5 10 GHz ∼ − ∼ − ∼ − gap f 106 GHz Coherence 600 µs 1 35 s 1 ms 1 s 5 20 µs ∼ ∼ − ∼ − ∼ − time T2 Coupling 25 MHz 10 kHz 100 Hz 150 MHz ∼ ∼ ∼ − ∼ rate g/2π 1 MHz

Table 1.1: Important properties for diﬀerent systems that can be used as qubits [49, 57, 80, 81, 83, 121].

At present there is considerable interest in the development of hybrid approaches to quantum information processing. These involve combining two or more distinct quantum systems, where, for example, one is being employed as the quantum memory, i.e. a quibit, and coupled to another quantum system, i.e. a resonator, to perform gate operations, in order to take advantage of the most favorable aspects of

23 Figure 1.1: Overview of hybrid quantum systems. Each individual system is positioned according to its excitation frequency and coherence time, with possible couplings between them indicated by arrows. From [65]. each. These may include, for example, coherence times, scalability or the timescales on which states can be prepared or manipulated. Within the last decade, hybrid approaches to quantum information processing have been proposed and realized in various ways. One such proposal is the use of ensembles of cold polar molecules acting as a long-lived quantum memory coupled to a circuit QED setup consisting of a superconducting cooper pair box strongly coupled to a stripline resonator [94]. Another proposal is the magnetic coupling of a Bose-Einstein condensate of rubidium atoms to the nanomechanical motion of a cantilever [110]. Examples

24 for the experimental realization of strongly coupled hybrid quantum systems are the coupling of electron spin ensembles to a superconducting transmission-line [100] as well as the emergence of circuit quantum acous- todynamics where a superconducting qubit has been coupled to trapped on-chip acoustic wavepackets [76].

Optical Coplanar wave- Si Nanobeam cavity guide resonator Mechanical oscillator Frequency 105 GHz 2 8 GHz 4 GHz ∼ ∼ − ∼ range Coupling strength 100 MHz > 300 MHz 1 MHz ∼ ∼ g/2π Quality > 106 105 108 105 ∼ − ∼ factor Q

Table 1.2: Important properties for diﬀerent types of resonators [57, 42, 81, 63, 25].

In this context the work described in this thesis is related to the investigation of chip-based approaches to exploiting Rydberg atoms in hybrid quantum information processing [122, 98, 72, 89, 88]. The long-term objetives of this work are to coherently couple gas-phase atoms in highly excited Ry- dberg states to solid-state superconducting qubits. The coupling between the two systems will be mediated, resonantly or near-resonantly through coplanar superconducting microwave resonators. Because of the possibil-

25 ity of implementing fast quantum gates (g/2π 100 MHz) when using ≈ superconducting qubits and the inherent scalability of chip-based structures [58], superconducting microwave circuits appear to be ideal solid- state candidates for the realization of quantum approaches to computation and simulation. However, superconducting qubits suﬀer from comparitively short coherence times (τ 10 µs). They are therefore not ideally ≈ suited as long coherence-time quantum memories. With this in mind eﬀorts are being directed toward the development of hybrid approaches to quantum information processing in which rapid state manipulation could be performed in these solid-state circuits while a gas-phase sample of atoms, molecules or ions is used as a long coherence-time memory [94]. From this perspective, atoms in highly excited Rydberg states are well suited to act as the gas-phase component of such a hybrid system. Rydberg states can exhibit comparitively long coherence times ( 100 ms) and because of ≈ their large electric dipole moments for transitions at microwave frequencies they are compatible with coupling to microwave circuits. A realization of such a hybrid system would open up possibilities for several new applications such as long-coherence-time quantum memories [104], or enabling the coupling between optical and microwave photons [13] in a controllable medium for microwave-to-optical conversion [38]. Other possiblities include using Rydberg atoms coupled to superconducting circuits for photon number state preparation as has been done in large three-dimensional cavities [111], and long distance quantum entanglement distribution [27] in spatially separated quantum circuits coupled via transmission lines. In approaching the goal of strongly coupling gas-phase Rydberg atoms to

26 superconducting microwave circuits, a chip-based architecture for com- prehensive control of the translational motion and internal quantum states of Rydberg atoms has recently been developed [66, 67, 51]. This architecture allows ensembles of Rydberg atoms to be accelerated, decelerated and transported in travelling electric traps located directly above coplanar electrode structures that are suitable for microwave transmission. The same electrode structures can act as coplanar microwave resonators and electrostatic traps for Rydberg atoms located above them [68]. In this thesis, theoretical studies are reported upon which those experimental developments will build. They include (1) studies of static electric dipole interactions within ensembles of Rydberg atoms which play an important role in the preservation of coherence over long timescales in these samples, (2) analysis of a scheme to use beams of Rydberg atoms to cool selected resonator modes, and (3) the extension of this cooling scheme to cool selected modes of other solid-state systems, such as nanomechanical oscillators. In Chapter 2 the theoretical description of hydrogen atoms in Rydberg states and its extension to helium atoms is reviewed. Then, the properties of Rydberg atoms in circular states are discussed, the general method to prepare atoms in such circular states is reviewed, and extended to the preparation of helium atoms in circular Rydberg states. In Chapter 3 the role of static electric dipole-dipole interactions between helium Ryd- berg atoms, which lead to mean-ﬁeld energy shifts in an atomic ensemble, is investigated. The general quantization procedure of superconducting transmission line resonators is then discussed in Chapter 4, and the open

27 quantum systems approach to quantifying the dissipation within these architectures when coupled to a thermal bath such as a coplanar waveguide used for measurements is then outlined. Chapter 5 deals with the general theory of quantum reservoir engineering and how it is applied to a beam of Rydberg atoms interacting with a transmission line resonator. Within this context an original cooling scheme for hybrid quantum systems with Rydberg atoms is presented and an analytic solution is given for the diagonal elements of the resonator density matrix. Chapter 6 improves upon this idea by introducing a numerical method that provides an even more eﬀective cooling mechanism as well as more insights due to the calculation of the full density matrix of the resonator. Chapter 7 introduces an additional hybrid quantum system including a nanomechanical oscillator, whose quantum states are controlled via quantum reservoir engineering by an atomic beam. Finally, in Chapter 8 open problems and the outlook of this research are discussed.

28 Chapter 2

Rydberg Atoms

Rydberg atoms are atoms excited to states with very high principal quantum number n. They have been of importance in the derivation of Bohr’s quantum mechanical orbital model of the electronic structure of atoms, and have played a key role in the development of quantum optics, quantum information and many-body physics due to the possibility of realiz- ing strong electric dipole coupling to single microwave photons [18] and strong dipole-dipole interactions [6] between pairs of Rydberg atoms. In this chapter some of the properties of Rydberg atoms, and methods of Rydberg state preparation relevant to the topic of this thesis are described.

2.1 Hydrogen atoms in Rydberg States

The simplest atoms to be theoretically described are the hydrogen and hydrogen-like atoms. These are two-particle quantum systems, which can be approximately described as consisting of a positively charged nucleus and one electron. The electrostatic interaction between these two particles

29 is expressed by the Coloumb potential

Ze2 V(r) = , (2.1) −4π0r where the charge of the nucleus is Ze, e is the charge of the electron, 0 is the electric permittivity of the vacuum, and r is the distance between the electron and the nucleus. The wavefunction of the electron Ψ can be described via the Schrödinger equation [28] ! ~2 Ze2 2 Ψ = EΨ, (2.2) −2µ∇ − 4π0r where ~ is the reduced Planck constant, and µ = meM is the reduced me+M mass of the electron-nucleus system with me and M being the mass of the electron and nucleus, respectively. The energy of the electronic eigenstates is denoted by E. Switching from Cartesian coordinates (x, y, z) to spherical coordinates q r = x2 + y2 + z2, (2.3)

z θ = arccos , (2.4) r y φ = arctan , (2.5) x

Eq.(2.2) can be seperated1 into radial and angular components. Hence, the wavefunction is factorizable as [11]

Ψ(r, θ, φ) = R(r) Y(θ, φ). (2.6) ·

If both the radial and angular parts of the Schrödinger equation are equal but with opposite signs, they cancel each other, leading to a nontrivial

2 1The Laplace operator transforms as 2 = 1 ∂ r2 ∂ + 1 ∂ sin θ ∂ + 1 ∂ ∇ r2 ∂r ∂r r2 sin θ ∂θ ∂θ r2 sin2 θ ∂ϕ2 30 solution. By deﬁning this separation constant [45] as `(` + 1), two self- consistent eigensystems follow:

Angular equation • " ! # 1 ∂ ∂ 1 ∂2 sin θ + Y(θ, φ) = `(` + 1). Y(θ, φ) sin θ ∂θ ∂θ Y(θ, φ) sin2 θ ∂φ2 − (2.7) The solutions to this equation are the fatorizable normalized spherical

harmonic functions Y`,m(θ, φ) = f`,m(θ)gm(φ), where the azimuthal quantum number m is introduced as another separation constant analogous to the factorization of the electron wavefuntion above. The generating function for the spherical harmonics is r m (2` + 1)(` m)! imφ Y`,m(θ, φ) = ( 1) − P`,m(cos θ)e − 4π(` + m)! (2.8)

= Y`,∗ m(θ, φ), −

with m 0, and where P m(cos θ) are the associated Legendre poly- ≥ `, nomials.

Radial equation • " ! 2 2 ! # d 2 d 2µr Ze r + 2 E + `(` + 1) R(r) = 0. (2.9) dr dr ~ 4π0r −

By substituting y(r) = rR(r), one arrives at a form of a well known equation whose orthonormal set of analytical solutions is related to

a the associated Laguerre functions Lb(x). The solution for the eigenstates is [44] s !3 !` ! Zµr 2Zµ (n ` 1)! 2Zµr 2`+1 2Zµr − nmea0 Rn,`(r) = − − 3 e Ln ` 1 nmea0 2n[(n + `)!] nmea0 − − nmea0 (2.10)

31 2 2 where a0 = 4π0~ /(mee ) is the Bohr radius and the bound states are categorized by the principal quantum number n = 1, 2, 3,... and the angular momentum quantum number 0 ` n 1. ≤ ≤ −

The eigenenergies follow from the solution to the radial equation and give the Rydberg formula Rhc 13.6 eV En = = − (2.11) − n2 n2 where the Rydberg constant for an electron bound to an inﬁnitely heavy positive point charge can be expressed in terms of the fundamental con- stants Z2e4m Z2~2 R = e = (2.12) hc 2 2 2 2 16π 0~ a0me Several properties of low-` Rydberg states follow from this treatment of the structure of the hydrogen atom. Especially interesting are the scaling laws of the atomic properties with the principal quantum number n. For higher Rydberg states, i.e., large n, the binding energy of the electron decreases

2 with n− . The energy diﬀerence between adjacent eigenstates decreases

3 by n− . Hence, the radiative lifetime of such transitions increases with n3. The large electric dipole transition moments, ~p, for transitions where

2 ∆n = n0 n = 1, scale as p n0`0 e zˆ n` n . This is a consequence of − ± ∝ h | · | i ∝ the large orbital radius of Rydberg states, which scales with n2. All these properties mean that atoms in high-n Rydberg states are well suited to applications in quantum information processing [36].

32 2.2 Helium atoms in Rydberg states

For helium atoms where one electron is excited to a high energy state the Rydberg formula Eq. (2.11) needs to be adjusted by introducing the quantum defect δn`. This empirical parameter accounts for the fact that the electron excited to a Rydberg state interacts with what can be viewed as an ion core whose nuclear charge is screened by the surrounding second electron, as can be seen in Fig. 2.1. Consequently, the Coloumb potential of the ion core is perturbed as it is no longer spherically symmetric. It is also possible that the inner electron is polarized by the outer electron in the Rydberg state. Therefore, the diﬀerence in binding energy for the hydrogen case can be accounted for by replacing the principal quantum number n by an eﬀective quantum number [36] n∗ = n δn . The adjusted − ` Rydberg formula for helium is

RHe hc En = 2 (2.13) −(n δn ) − ` R µHe where R = ∞ is the Rydberg constant for the helium atom. As can be He me inferred from Fig. (2.2), the deviation from our model of the ion core and the inner electron as one composite ion core with charge Z = +1 is bigger for the outer electron in Rydberg states closer to the positive core. Thus, the quantum defect, δn`, decreases for Rydberg states with higher quantum numbers n and `.

2.3 Atoms in Circular Rydberg States

Rydberg states which are of particular interest as qubit states in hybrid cavity QED are those which possess maximum angular momentum and

33 Electron in high n, ℓ Rydberg state

Inner electron and ion core

Figure 2.1: The helium atom in a high n Rydberg state. The outer electron "sees" the ion core and inner electron as one central ﬁeld with charge Z = +1. The Coloumb potential associated with the He2+ nucleus is screened by the inner electron.

magnetic quantum numbers ` and m`, respectively, for a particular value of n. Such states are called circular Rydberg states and exhibit unique properties. The first of these are their long radiative lifetimes. Since the Einstein A coefficients describing the transition rate from an initial state i to a final state f , i.e., the decay rate of an excited state, can be written as a sum of the Einstein A coefficients of the decay into all possible final states. The lifetime of the excited state is then the inverse of this decay rate, and

34 Figure 2.2: The effect of the quantum defects δn,` on the binding energy of the atomic states for non-hydrogenic atoms compared to the hydrogen atom. From [53]. thus   1 2ω3 − 1 X i f `max   ~ 2 τ = P =  3 p f i  , (2.14) A f i  30~c 2ì + 1| |  f  f  where A f i is proportional to the cube of the energy (frequency) difference between the initial and final states, and the square of the transition dipole moment [33]. In the case of circular Rydberg states, the excited state has ` = m = n 1. ` − Due to the electric dipole selection rules allowing up to first order only transitions satisfying `0 = ` 1 and m0 = m or m0 = m 1, it follows that − ` ` ` ` ±

35 decay can only occur to ﬁnal states `0 = n 2. Consequently, the initial − excited circular state can only decay to one unique ﬁnal state, n0 = n 1, − `0 = n 2, m0 = m 1. Therefore, circular Rydberg states are a very − | `| | `| − good approximation to 2-level systems. Within the dipole approximation this quasi two-level characterization only breaks down on long timescales since the lower n0 = n 1 circular state can itself decay to the next higher- − lying circular state with n00 = n 2. However, in an electromagnetic − environment that is appropriately engineered such that no vacuum mode exists at the frequency of the n 1 n 2 circular-state to circular- − → − state transition, this additional decay channel can be eliminated [59]. The transition energy between the two states making up the quasi two-level

3 system is En,n 1 n− , while their electron dipole transition moment scales − ∝ as n 1, n 2 ~r n, n 1 n2. Therefore, it follows for the radiative lifetime h − − | | − i ∼ of circular Rydberg states

1 5 τ f i 3 3 2 2 = n . (2.15) ∝ (n− ) (n )

Hence, circular Rydberg states exhibit greatly enhanced radiative lifetimes compared to low-` states with the same values of n. The properties discussed above are listed in Tab. (2.1) for two examples of Rydberg states of Helium, showing an increased ﬂuorescence lifetime in the case of circular Rydberg states when compared to states with low orbital angular momentum.

Circular Rydberg states cannot be directly photoexcited from an atomic ground state because of the selection rules for electric dipole transitions. However, two methods have been developed and implemented experi-

36 Table 2.1: Comparison between some of the properties of low- and high- angular-momentum Rydberg states. mentally to prepare them. The first of these involves adiabatic microwave transfer from an initially laser photoexcited low-` (low-m`) Rydberg state with the absorption of approximately n microwave photons [60, 69]. The second method to produce circular Rydberg states is known as the crossed-fields method, and was first proposed by Delande and Gay [29]. By applying a constant magnetic field crossed perpendicularly with an ad- justable electric field, one is able to very efficiently prepare atoms in circular

Rydberg states by laser photoexcitation to approppriate `- and m`-mixed states which then evolve into circular states when the electric ﬁeld strength is adiabatically reduced towards zero. The Hamiltonian associated with a single Rydberg electron in weak crossed electric F~ = (Fx, 0, 0) and magnetic

B~ = (0, 0, Bz) ﬁelds can be written in atomic units to ﬁrst order as

2 p 1 Bz m` = + + Fx x, (2.16) H − 2 − r 2 | {z } | {z } c W H where is the Coulomb-interaction Hamiltonian and W the pertubation Hc due to the applied ﬁelds. The crossed-ﬁelds approach to the preparation of

37 circular states is based on the SO(4) symmetry of the Coulomb interaction,

c, with the generators Łij [87], where H

Łij = εijk Lk, · (2.17) Łi4 = Ai.

Here εijk is the fully antisymmetric tensor, ~L is the angular momentum vector and A~ is the Runge-Lenz vector. One suitable choice of eigenbasis for this Hamiltonian is the subgroup chain SO(4) SO(3) SO(2)m, ⊃ λ ⊃ which preserves the rotational invariance in real space. Their associated 2 eigenfunctions ~λ and λz, derived from the operator ~λ = (Ł14, Ł24, Ł12), are solutions to the Coulomb problem. The corresponding energy eigenvalues of the hydrogen atom in atomic units are

1 E(0) = (2.18) n −2n2

The perturbation due to the applied ﬁelds W can be diagonalized for each n atomic shell. After replacing the operator x = (3/2)n Ax [87], one obtains − ·

Wn = ωLLz + ωSAx q = ω2 + ω2~λ u~, (2.19) L S ·

Here ω = Bzm /2 and ω = ( 3/2)nFx are the Larmor and Stark frequencies L ` S − in atomic units, and u~ = (sin α, 0, cos α) can be interpreted as a rotation of

1 the operator ~λ in the xz-plane by the angle α = tan− (ωS/ωL).

From Eq. (2.18) and Eq. (2.19) it can be seen that the eigenfunctions of c, H (~λ)2, and W commute, so they can be chosen as the eigenfunctions of the total Hamiltonian, whose eigenenergies are q 1 2 2 En = + k ω + ω , (2.20) −2n2 L S 38 Figure 2.3: Adiabatically switching oﬀ the electric ﬁeld, F~, leads to a rotation of the vector operator ~λ. This is equivalent to the evolution of the upper state’s parabolic ellipse, with angular momentum m = 0, to a

circular trajectory with m = Lz = n 1. − where k is the eigenvalue of the (λ u~)-operator. Circular Rydberg states · can therefore be prepared by starting out in the Stark limit ωS ωL, optically exciting the outermost Stark state with maximum eccentricity k = n 1 = λx = Ax, as illustrated in Fig. (2.3). This is possible because max − the projection of the angular momentum vector in the direction of the applied electric field is Lx = 0 = m. In this case the vector ~λ is directed along the x-axis (electric field axis), because α π , which is the angle ≈ 2 between the z-axis and ~λ. By slowly switching off the electric field, the system evolves adiabatically to the limit in which ω ω , i.e., α 0, and L S ≈ a circular Rydberg state λ = Lz = n 1 = m , while remaining in the − max upper sublevel, k = n 1, because the magnitude of the vector operator max − ~λ = n 1 is conserved since the upper sublevel is a coherent state of the | − i SO(3)λ subgroup. Using this crossed-fields method a circular Rydberg state results, as the electron evolves into the state with k , m . Even though this method | max maxi is described here for the case of the hydrogen atom, as long as the upper

39 state Lx = 0 can be selectively prepared by either laser photoexcitation from the ground state or by optical-microwave double resonance methods, it can be used to eﬃciently produce circular Rydberg states of any atomic species, such as helium [127]. It has also been applied in the preparation of Rydberg states in hydrogen [73], rubidium [16], cesium [59], and other atoms [82, 26, 5].

2.3.1 Helium Atoms in Circular Rydberg States

Helium atoms are well suited to hybrid quantum information processing with Rydberg atoms and transmission line resonators because of their low adsorption rates onto cryogenic chip surfaces which in turn minimizes the build-up of time-varying stray electric fields [109]. In these experiments it is desirable to operate the superconducting circuits at frequencies below 20 GHz. At such frequencies resonators with higher quality factors ∼ can be realized. To achieve this therefore requires atoms in circular Rydberg states with n 70. However, the preparation of circular Rydberg states ≥ with these high values of n is particular challenging due to the high sensitivity of the photoexcitation phase of the crossed-fields method to stray electric fields. For non-hydrogenic atoms such as helium the standard approach to circular state preparation using the crossed-fields method is to initially excite the outermost positively shifted Stark state with m` = 0, with respect to the electron field quantization axis, at the first avoided crossing of this state with a low-` state at the Inglis-Teller limit. The advantage of this approach is that at this avoided crossing the target outer Stark state can possess a significant fraction of the character of the low-` state, mak-

40 Figure 2.4: The energy level diagrams for steps 2-4 in the preparation process of circular Rydberg states for n = 70 in Helium as described in the text. From [80]. ing laser photoexcitation more efficient. However, for high values of n, the avoided crossings of these sublevels can be so small that the transitions to the individual states are not resolvable during the laser photoexcitation. To overcome this problem, the usual crossed-fields method, introduced in the previous section, can be modified [80]. Regarding notation, the circular states for specific values of n are denoted

41 here by nc for values of m` = (n 1). To prepare the helium atoms | ±i ± − in high-n circular Rydberg states, e.g. the 70c circular state, a resonant | +i two-photon laser excitation via an intermediate level is first performed in a weak electric field to transfer atoms from the metastable triplet 2s state | i to the 73s Rydberg state. In the second step of the process illustrated in | i Fig. (2.4) the electric field strength is adiabatically increased to polarize the

73s0 atoms, where the prime indicates an `-mixed Stark state with in a | i nonzero electric field, such that they have a static electric dipole moment of similar magnitude and orientation as the target outer Stark state with n = 70, i.e., the 70c0 state. This is achieved in a field just beyond the first | +i avoided crossing of the 73s state in the Stark map of the triplet Rydberg | i states of He. Due to the large overlap of the wavefunctions of this state and the 70c0 state, the corresponding large electric dipole transition moment | +i allows for fast and efficient population transfer to the 70c0 state by apply- | +i ing a resonant microwave pulse. Once the 70c0 state has been prepared, | +i the fourth and last step of the circular state preparation process is to slowly reduce the electric field strength so that the atoms evolve adiabatically in the presence of the magnetic field into the final 70c circular state. The | +i transition between the 70c and 71c circular Rydberg states possesses | +i | +i an electric dipole transition moment of 3477ea0, allowing for single photon coupling strengths on the order of g/(2π) 5 MHz [80] to two-dimensional ≈ superconducting microwave resonators. Circular states, such as these, are particularly well suited for hybrid cavity QED because they possess long coherence times ( 100 ms), low sensi- ∼ tivity to stray electric fields and possess large electric dipole transition

42 moments and transition frequencies in the microwave regime. Therefore, in the theoretical studies of involving Rydberg-atom–resonator coupling in Chapters 5, 6 and 7 of this thesis, the interaction of these high-n circular Rydberg states will be considered.

43 Chapter 3

Electrostatic Dipole Interactions in Polarized Rydberg Gases

The preparation of circular Rydberg states using the crossed fields method described in Chapter 2 requires the initial laser photoexcitation of Rydberg states with strong linear Stark shifts, and hence large static electric dipole moments. These states are then adiabatically converted into circular states in a zero electric field. To achieve quantum-state–selective preparation of a large number of circular Rydberg atoms the applied fields must be care- fully controlled and interactions between the excited atoms in each phase of the preparation process must be minimized. When the circular preparation process is complete the dominant type of interaction between pairs of

6 excited atoms is the van der Waals interaction that scales with R− , where R is the distance between the atoms. However, the large static electric dipole moments of the Rydberg-Stark states initially laser photoexcited interact through the stronger static electric dipole-dipole interactions, which

3 depend on R− . A wide range of studies, both theoretical and experimen-

44 tal, have been carried out on van der Waals and resonant dipole-dipole interactions between Rydberg atoms [35, 9, 34]. These interactions, and the associated excitation blockade mechanisms, are exploited in, e.g., the entanglement of pairs of atoms [91, 118], and in applications of Rydberg atoms in quantum simulation [61, 79]. However, there has been little com- parable study on the large static electric dipole interactions relevant to the circular state preparation processes discussed in Chapter 2. For this reason, and with the aim of elucidating the role of such interactions in Rydberg- Stark deceleration and trapping experiments, described in this chapter are the results of experiments and calculations of eﬀects of static electric dipole interactions in dense gases of strongly polarized helium Rydberg atoms. The work described in this chapter was published in Zhelyazkova, Jirschik, and Hogan, Physical Review A 94.5: 053418 (2016) [128]. The text here follows closely the content of the published article, with adaptations to ﬁt the structure of this thesis. The experiments described in Sections 3.1 and 3.2 were performed by Dr. Valentina Zhelyazkova while I carried out the calculations described in Section 3.3. Dr. Valentina Zhelyazkova and I both contributed to the preparation of the text of the published article.

The static electric dipole moments of Rydberg atoms and molecules can exceed 10000 D for states with values of n 50 and scale with n2. These ≥ dipole moments give rise to strong linear Stark shifts in external electric ﬁelds [36], and allow for control of the translational motion of a wide range of neutral atoms and molecules [54], composed of matter [123, 114, 52, 55] and antimatter [30], using inhomogeneous electric ﬁelds. When prepared

45 in states with these large dipole moments, gases of Rydberg atoms can also exhibit strong electrostatic dipole-dipole interactions [37, 125, 117]. These inter-particle interactions cause energy shifts of the Rydberg states, and, as demonstrated here, can modify the dielectric properties of the gases. The observation, and spectroscopic characterization, of the effects of these electrostatic interactions opens opportunities for using beams of Rydberg atoms or molecules as model systems with which to study many-body processes [23], including, e.g., resonant energy transfer [103], or surface ionization [50, 70, 40]. It is also of importance in the elucidation of effects of dipole interactions in Rydberg-Stark deceleration and trapping experiments [101], in experiments involving long-range Rydberg molecules [43, 10] possess- ing large static electric dipole moments [14], and in applications of Rydberg states as microscopic antennas for the detection of low-frequency electric fields [125, 78]. Microwave spectroscopy of the effects of van der Waals interactions [95, 86, 106], and resonant dipole-dipole interactions [2, 85] in cold Rydberg gases have been reported previously and provided important insights into these interacting few-particle systems. The electrostatic interactions that occur between Rydberg atoms in the outermost Stark states with the largest electric dipole moments, µ (3/2)n2ea , represent the extreme case of max ' 0 the dipole-dipole interaction for each value of n. The interaction potential, V , between a pair of atoms with electric dipole moments, µ = µ~ and dd 1 | 1| µ = µ~ , aligned with an electric field, F~, can be expressed as [37] 2 | 2|

µ1µ2 2 Vdd = 3 1 3 cos θ , (3.1) 4π0R − where R = R~ is the inter-atomic distance, θ is the angle between R~ and F~, | | 46 and 0 is the vacuum permittivity. For an isotropic, aligned ensemble of dipoles the average interaction energy is non zero, and leads to mean-ﬁeld energy-level shifts within the ensemble. The microwave spectroscopic studies reported here represent a direct measurement of these mean-ﬁeld shifts in gases of helium (He) excited to Rydberg states with n = 70 and electric dipole moments up to µ1 =

µ70 = 12 250 D. In the highest density regions of these gases, the dielectric properties differ from those in free space and give rise to changes in the microwave spectra which reflect the emergence of macroscopic electrical properties of the medium. This work is complimentary to recent studies of effects of electrostatic dipole interactions on particle motion in laser cooled samples of rubidium [108], and observations of optical bistability arising from mean-field shifts in gases of Rydberg atoms interacting via resonant dipole interactions [22].

3.1 Experiment

The experiments were performed in pulsed supersonic beams of metastable helium. The He atoms in these beams had a mean longitudinal speed of

1 1 2000 m s− with a spread of 50 m s− [126]. The atoms were prepared in ∼ ± 3 the metastable 1s2s S1 level in an electric discharge at the exit of a pulsed valve. This valve has operated at a repetition rate of 50 Hz [46]. After col- limation by a 2 mm diameter skimmer and the deﬂection of stray ions pro- duced in the discharge, the beam entered between a parallel pair of 70 mm 70 mm copper electrodes. The electrodes were separated by 8.4 mm × in the z dimension, as indicated in Fig. 3.1(a). At the mid-point between

47 (a) MCP V1 v He* z F 8.4 mm z 20 mm V2 x y (b) n = 70 n = 72 signal - e Time Ionization electric field

11 µs ~1 µs

Microwave pulse

∆tµ ~ 1 µs

Polarization electric field

Photoexcitation 200 ns 5 µs pulse

3 µs Time

Figure 3.1: (a) Schematic diagram of the photoexcitation and detection region of the experimental apparatus. The position of laser photoexcitation and the approximate spatial extent of the Rydberg atom ensemble in the y dimension during microwave interrogation are indicated by the red circle and dashed rectangle, respectively. (b) Sequence of events in each experimental cycle, including laser photoexcitation, Rydberg state polarization, microwave interrogation, and state-selective electric ﬁeld ionization (not to scale). From [128].

48 these electrodes the atomic beam was crossed at right-angles by two co- propagating, frequency stabilized, cw laser beams. These lasers were operated at wavelengths of λUV = 388.975 nm and λIR = 785.946 nm to excite the metastable atoms to Rydberg states by the 1s2s 3S 1s3p 3P 1s70s 3S 1 → 2 → 1 two-photon excitation scheme. The laser beams were focussed to full- width-at-half-maximum (FWHM) beam waists of 100 µm. By applying ∼ a pulsed potential, V1, to the upper electrode in Fig. 3.1(a) to bring the atoms in the photoexcitation region into resonance with the lasers for 3 µs [see Fig. 3.1(b)], 6 mm-long ensembles of Rydberg atoms were generated in each cycle of the experiment. After photoexcitation, the electric ﬁeld was switched again to polar-

3 ize the 1s70s S1 atoms [see Fig. 3.1(b)]. Interactions within the excited ensemble were then probed by microwave spectroscopy on the single photon transition at 37 GHz between the states that evolve adiabatically to ∼ 3 3 the 1s70s S1 and 1s72s S1 levels in zero electric field. These states have quantum defects of δ δ 0.296 664 [32] and exhibit quadratic Stark 70s ' 72s ' energy shifts in weak electric fields. Because the laser photoexcitation process was separated in time from microwave interrogation, the Rydberg atom number density, nRy, could be adjusted by controlling the IR laser intensity, without affecting any other experimental parameters. After trav- eling 20 mm from the position of laser photoexcitation, the Rydberg atoms were detected by state-selective ramped-electric-field ionization beneath a 3 mm-diameter aperture in the upper electrode [see Fig. 3.1(a)]. This ionization step in the experiments was carried out by applying a slowly- rising ionization potential, V2, to the lower electrode. The electrons from

49 -630 n = 72 72p -640 72s (i) (ii) (iii)

-650 n = 71 71p -660 71s Energy / h (GHz)

-670 n = 70 70p 70s -680

0 0.2 0.4 0.6 0.8 1.0 1.2 Electric field (V/cm)

Figure 3.2: Stark structure of the triplet m` = 0 Rydberg states of He with values of n ranging from 70 to 72. Experiments were performed following laser photoexcitation of the 70s state in zero electric ﬁeld and the subsequent polarization of the excited ensemble of atoms in ﬁelds

1 1 1 of (i) 18 mV cm− , (ii) 358 mV cm− , and (iii) 714 mV cm− as indicated. From [128].

50 the ionized Rydberg atoms were detected on a microchannel plate (MCP) detector located above this electrode. The rise time of the pulsed ionization electric ﬁeld was selected to separate in time the electron signal from the n = 70 and n = 72 states.

3.2 Results

The mean-ﬁeld energy-level shifts in the ensembles of polarized Rydberg gases were observed directly by microwave spectroscopy. To achieve this, reference measurements were ﬁrst made in which the polarizing electric

1 1 field was switched after photoexcitation from 0 1 mV cm− to 18 mV cm− , ± 3 inducing electric dipole moments of µ70 = 380 D in the 1s70s S1 atoms. Here, and in the following, the dipole moments referred to represent the absolute value of the first derivative of the Stark energy shift with respect to the electric field. The n = 70 n = 72 transition (µ = 470 D), → 72 indicated by the dashed line labelled (i) in Fig. 3.2, was then driven by a 1 µs-long microwave pulse [see Fig. 3.1(b)]. The corresponding Fourier- transform-limited spectrum of the integrated electron signal associated with the n = 72 state is displayed in Fig. 3.3(a-i) (continuous curves). In order to identify and characterize the mean-field energy-level shifts in the ensembles of excited atoms measurements were performed at two Rydberg atom densities. The low (high) density data were recorded following the excitation of NRy (16 NRy) Rydberg atoms, and are displayed in blue (red) in the figure. For the small electric dipole moments induced in this case, no density-dependent energy-level shifts were observed. To enhance the effects of electrostatic dipole interactions within the

51 (a) (b) (iii) (iii) 3.0 3.0 µ = 14050 D Low High ) 72 )

µ70 = 12250 D density density 2.5 2.5

2.0 2.0 (ii) (ii) µ = 8650 D Low High 1.5 72 1.5 density density µ70 = 7250 D

1.0 1.0 (i) (i)

0.5 µ72 = 470 D Experiment 0.5 Experiment

µ70 = 380 D Calculation Calculation Integrated electron signal (arb. units Integrated electron signal (arb. units 0.0 0.0

-15 -10 -5 0 +5 +10 +15 -15 -10 -5 0 +5 +10 +15 Relative frequency (MHz) Relative frequency (MHz)

Figure 3.3: Experimentally recorded (continuous curves) and calculated (dashed curves) spectra of transitions between the Stark states that evolve

3 3 adiabatically to the 1s70s S1 and 1s72s S1 levels in zero electric ﬁeld. The experiments were performed at low (blue curves) and high (red curves) Rydberg atom number densities, and spectra were recorded in polarizing

1 1 1 electric ﬁelds of (i) 18 mV cm− , (ii) 358 mV cm− , and (iii) 714 mV cm− , for which µ70 = 380 D, 7250 D, and 12250 D (µ72 = 470 D, 8650 D, and 14050 D), respectively. The calculated spectra are presented (a) without, and (b) with eﬀects of the local polarization of the medium on the macroscopic dielectric properties accounted for. The relative microwave frequencies on the horizontal axes are displayed with respect to the transition frequencies [(i) 37.2413 GHz, (ii) 37.1068 GHz, and (iii) 36.8094 GHz] recorded at low Rydberg atom number density. From [128].

52 Rydberg gases, subsequent spectra were recorded with further polariza-

1 1 tion. Increasing the electric ﬁeld to 358 mV cm− (714 mV cm− ) resulted in induced electric dipole moments of µ70 = 7250 D (µ70 = 12250 D), and

µ72 = 8650 D (µ72 = 14050 D). In the spectra recorded in these fields, and displayed in Fig. 3.3(a-ii) and (a-iii), respectively, a significant dependence on the Rydberg atom density was observed. Comparison of the spectra in Fig. 3.3(a-ii) and (a-iii) with those in Fig. 3.3(a-i) reveals four notable features. When the atoms are more strongly polarized the spectral profiles: (1) become significantly broader than those in Fig. 3.3(a-i), even at low density; (2) exhibit density-dependent frequency shifts; (3) become asym- metric at high density with a sharp cut-off in intensity at higher microwave transition frequencies; and (4) display signatures of spectral narrowing at high density, in particular in Fig. 3.3(a-iii).

3.3 Simulations

To aid in the interpretation of the spectra, Monte Carlo calculations were performed in which the electrostatic dipole-dipole interactions within the many-particle system were treated. In these calculations, ensembles of Rydberg atoms were generated with randomly assigned positions. These ensembles had Gaussian spatial distributions with a FWHM of 100 µm (50 µm) in the x (z) dimension, and flat top distributions with a length of 3 mm in the y dimension. These distributions represent the spatial intensity profile of the laser beams in the z dimension, the spatial distribution of atoms with Doppler shifts within the spectral width (∆ν 15 MHz) FWHM ' of the 1s2s 3S 1s3p 3P transition in the x dimension, and half of the 1 → 2 53 length of the ensemble of atoms excited in the y dimension. For each atom in the calculation the difference between the sum of the dipole-dipole interaction energies with all other atoms when it was (1) in the n = 70 state, i.e., µ1 = µ2 = µ70, and (2) in the n = 72 state, i.e., µ1 = µ72, and µ2 = µ70, was determined. To account for the sharp cut-off in the spectral profiles at high transition frequencies in the data recorded at high density in Fig. 3.3(a-ii) and (a-iii), it was necessary to impose a lower limit,

Rmin, on the nearest-neighbor spacing between pairs of Rydberg atoms in the calculations. This minimum inter-atomic spacing is a consequence of the expansion of the pulsed supersonic beams as they propagate from the valve to the photoexcitation region in the experiments, a distance of 210 mm, and reflects the collision-free environment characteristic of these beams [102, 71]. From a global fit to all of the experimental data in Fig. 3.3, carried out by calculating and subsequently minimizing the root mean- squared error between the experimental data and the models obtained for different values of Rmin, the most appropriate value of Rmin was found to be 11.5 µm.

3 We note that because the C6, van der Waals coefficient of the 1s70s S1 level in He is 9.9 GHz µm6, blockade at laser photoexcitation does not ∼ play a major role at the laser resolution in the experiments for atoms separated by more then 3.5 µm and therefore does not dominate this value ∼ of R . Under these conditions, the distribution of n = 70 n = 72 tran- min → sition frequencies was determined for each set of electric dipole moments (i.e., for each electric field strength) for atoms located within 0.75 mm of ± the center of the ensemble in the y-dimension to avoid edge effects. The

54 resulting data were convoluted with a Gaussian spectral function with a FWHM of 1 MHz corresponding approximately to the Fourier transform of the microwave pulses. When the polarization of the individual atoms is increased they become more sensitive to low-frequency electrical noise [126]. This sensitivity led to the broadening of the resonances in the low-density regime in Fig. 3.3(a- ii) and (a-iii). The effect of this spectral broadening was introduced in the calculations as a perturbing electric field which caused a relative shift in the energy of the n = 70 and n = 72 Stark states, proportional to their electric dipole moments. The spectra calculated following the generation of low density samples containing NRy = 2 813 atoms and including the contributions from this electric field noise are displayed in Fig. 3.3(a) (blue dashed curves). Comparing the experimental data with the results of these calculations indicates that the broadening observed was commensurate

1 with white noise with a root-mean-square amplitude of Fnoise = 2 mV cm− . In the more polarized gases, increasing the Rydberg atom density is seen to shift the resonant microwave transition frequencies by +3 MHz ∼ and +5 MHz in Fig. 3.3(a-ii) and (a-iii), respectively, from those recorded ∼ at low density. These density-dependent changes indicate that atom-atom interactions dominate the spectral broadening caused by Fnoise. The mean- ﬁeld shifts in transition frequency were found to agree with those seen in the results of the calculations upon increasing the number of excited atoms by a factor of 16, matching the experiments, to NRy = 45 000 [red dashed curves in Fig. 3.3(a)]. The positive shifts in transition frequency with increasing density are a consequence of the predominantly repulsive

55 electrostatic dipole interactions within the elongated ensemble of Rydberg atoms in the experiments, and the larger energy shift of the more polar, n = 72 state under these conditions. However, the spectral narrowing observed at high density in the experimental data is not seen in the calculations in Fig. 3.3(a). To account for this it was necessary to consider the contribution of the local polarization of the Rydberg gases on their dielectric properties [84]. In a simple model of the dielectric Rydberg gas, the local polarization Ploc = nloc µ70 (nloc is the Rydberg atom number density obtained in the numerical calculation within a sphere of radius 25 µm surrounding each atom) can be considered to shield each atom from the laboratory electrical noise, reducing it to F = max( 0, F S P / ), where S is a constant factor that accounts loc { noise − κ loc 0} κ for the geometry of the ensemble of atoms in the experiments. This shape factor is a ﬁtting parameter associated with the proportion of local Rydberg atoms aﬀected by the dielectric screening of the noise, which varies with

Rmin and accounts for the arbitrary choice of the radius of the sphere in which the local atom density is calculated. For Sκ = 0.1, there is excellent agreement between the results of the calculations and the experimental data as can be seen in Fig. 3.3(b). This indicates the emergence of macroscopic electrical properties of the Rydberg gas when strongly polarized. In the experiments, free He+ ions, generated by photoionization of Ryd- berg atoms, would give rise to spectral broadening, or, if they increase the strength of the local electric ﬁelds within the ensembles of atoms, shifts of the spectral features toward lower transition frequencies (see Fig. 3.2). They are not expected to contribute to spectral narrowing such as that

56 4.0 (d) Experiment 3.5 Calculation 8 -3 nRy ≈ 3.3 × 10 cm 3.0 (c)

8 -3 2.5 nRy ≈ 2.8 × 10 cm

2.0 (b) n ≈ 2.1 × 108 cm-3 1.5 Ry

1.0 (a) n ≈ 7.5 × 107 cm-3 0.5 Ry Integrated electron signal (arb. units)

0.0

-15 -10 -5 0 +5 +10 +15 Relative frequency (MHz)

Figure 3.4: Experimentally recorded (continuous curves) and calculated (dashed curves) dependence of the n = 70 n = 72 spectra of the most → strongly polarized ensemble of atoms, for which µ70 = 12 250 D and

µ72 = 14050 D, on the Rydberg atom density. From [128].

57 observed in Fig. 3.3(a,ii-iii) and (b,ii-iii). For these reasons, and because we do not observe free ions or electrons in the process of ionization of the Rydberg atoms, we conclude that they do not contribute signiﬁcantly to the experimental observations. Further comparisons of experimental and calculated spectra of the most polarized atoms [Fig. 3.3(b,iii)] over a range of Rydberg atom densities are presented in Fig. 3.4. From these data, it can be seen that the experimental spectra, recorded upon increasing NRy by factors of 5, 9.5 and 16 [Fig. 3.4(b), (c) and (d), respectively] from the initial low density case [Fig. 3.4(a)], agree well with the calculated spectra containing equivalent proportions of atoms, i.e., NRy = 2 813, 14 063, 26 719, and 45 000. From the results of the calculations the mean Rydberg atom number densities, nRy, within

7 3 the samples could be determined to range from n = 7.5 10 cm− to Ry × 8 3 3.3 10 cm− , as indicated. In these strongly polarized gases the labo- × ratory electric ﬁeld noise, F , is completely screened when S P / noise κ loc 0 ≥ 1 8 3 2 mV cm− , i.e., when n 4.5 10 cm− , and the absolute energy-level loc ≥ × shift of each atom is 20 MHz. ∼

3.4 Conclusions

In conclusion, we have carried out spectroscopic studies of mean-ﬁeld energy-level shifts in strongly polarized Rydberg gases by driving microwave transitions between Rydberg-Stark states with similar electric dipole moments. The resulting spectra yield detailed information on the spatial distributions of the atoms. Spectral narrowing observed at high number density is attributed to changes in the local dielectric properties

58 within the medium from those in free space, and reﬂects the macroscopic electrical properties of the atomic samples that emerge under these conditions. From these results, it is seen that for atoms prepared in states with static electric dipole moments on the order of 10000 D dipole-dipole interactions result in frequency shifts in excess of 10 MHz for atom-atom separations of / 10 µm. To ensure that such frequency shifts remain below this value, and hence, are not detrimental to quantum-state–selective circular state preparation the number densities of excited atoms in experiments

8 1 must be maintained below 10 cm− .

59 Chapter 4

Superconducting Transmission Line Resonator

In the development of hybrid approaches to quantum optics and quantum information processing superconducting circuits and qubits are uniquely scalable systems with high gate operation rates but short coherence times [115]. On the other hand, atoms in Rydberg states offer lower gate operation rates for similar transition frequencies but, being in the gas phase, can possess significantly longer coherence times [97]. The key component required for coupling these two qubits is a high quality microwave resonator. To maintain scalability, transmission line resonators offer particular advantages in such hybrid system setups. In this chapter the characteristic properties of superconducting transmission line resonators are described. These properties are then employed in the theoretical studies in the following chapters.

60 Figure 4.1: Simple LC-circuit electrical oscillator: (a) In order to derive a Lagrangian where the single degree of freedom is the charge, q, the usual sign convention for the flow of the current, I, from q to q can be − used. (b) The sign convention in the text, where the positive and negative signs of the charge, Q, have been flipped with respect to the current, leads to the magnetic flow as the single degree of freedom in the Lagrangian describing the LC-circuit. From [41].

4.1 Quantization of a 1D Transmission Line Res-

onator

A one-dimensional transmission line resonator can be modelled as a series of harmonic LC-circuits. This electric circuit element consists of an inductor with inductance L and a capacitor with capacitance C. The resonant circuit is described by the potential energy, EC, stored in the capacitor

CV2 E = , (4.1) C 2

61 where V is the voltage across the capacitor, and the kinetic energy stored in the magnetic field, EL, which is induced by the current I, flowing through the inductor 1 E = LI2. (4.2) L 2 The inductor current I = Q˙ is the rate of change of capacitor charge with time. The magnetic flux Φ can be expressed in terms of the applied voltage using Faraday’s law of inductance,

Z t Φ(x, t) = dτ V(x, τ). (4.3) 0

Considering the deﬁnition of the inductance, i.e., L = Φ/I, the energy stored in the capacitor has the form of a kinetic energy and the energy stored in the inductor is analogous to the potential energy of a particle in one-dimensional space. Consequently, the Lagrangian for the LC-circuit can be written in terms of the magnetic ﬂow as

1 2 1 2 LC = CΦ˙ Φ . (4.4) L 2 − 2L

The conjugate momentum to this single degree of freedom, Φ, is therefore

∂ L = CΦ˙ (4.5) ∂Φ˙ = Q, (4.6) and hence the capacitator charge Q. Thus, the Hamiltonian for the LC- circuit is derived as

HLC = QΦ˙ − L !2 (4.7) C dΦ 1 = + Φ2, 2 dt 2L

62 which is the electronic version of a single-mode harmonic oscillator. The analogous lumped circuit element model of a 1D transmission line is a series of linearly coupled LC-circuits. The Hamiltonian associated with the transmission line can therefore be written as a discretized sum of N inductively coupled LC-circuits

N !2 N X C dΦ X 1 2 HTL = + Φj Φj 1 . (4.8) 2 dt 2L − − j=1 j=2

This expression has the same form as a harmonic chain. In the continuum limit, a capacitance per unit length, c, and inductance per unit length, l, can be deﬁned. For a transmission line of length d, the Lagrangian can then be written as [12] Z d c 2 1 2 TL = dx (∂tΦ) (∂xΦ) , (4.9) L 0 2 − 2l which leads to the Euler-Lagrange equation ! d2 1 ∂2 Φ = 0. (4.10) dt2 − lc ∂x2

This is the equivalent of a wave equation with velocity v = 1 . In order √lc that the transmission line acts as a resonator which stores an electromagnetic ﬁeld in form of standing waves, series capacitances are implemented at each end of the transmission line to impose open-circuit boundary conditions such that

∂Φ ∂Φ = = 0. (4.11) ∂x x=0 ∂x x=d The boundary conditions are time-independent. Hence, the magnetic ﬂux ﬁeld can be factorized

X∞ Φ(x, t) = fn(t)φn(x) (4.12) n=0

63 where the amplitude fn(t) of each ﬁeld mode n depends only on time and

φn(x) are the normalized spatial eigenfunctions, whose derivatives vanish at the boundaries. These normal modes are expressed as r 2 φ (x) = cos (k x) (4.13) n L n

nπ where kn = with n N. The properties of these normal modes, i.e., that L ∈ Z L dxφn(x)φm(x) = δnm, (4.14) and hence Z L 2 dx∂xφn(x)∂xφm(x) = knδnm, (4.15) permit their use as an orthonormal basis in which to diagonalize the La- grangian in Eq. (4.9),

X∞ c h 2 2i = (∂t fn) ωn f , (4.16) LTL 2 − n n=0 where the wavevector is deﬁned as kn = ωn/v. This Lagrangian, and subsequently the Hamiltonian, constitute a sum of independent simple harmonic oscillator normal modes, where the momentum conjugate to the amplitude fn(t) is ∂ qn = L = c∂t fn. (4.17) ∂(∂t fn) This set of independent normal modes can be promoted to quantum op- h i erators by imposing the commutation relation qˆn, fˆm = i~δn m. Since the − , classical Hamiltonian, which follows from Eq. (4.16),

X∞ 1 c H = q + ω2 f 2 , (4.18) TL 2c n 2 n n n=0

64 has a form that resembles the free electromagnetic field, and its quantization can be performed in a similar fashion [116]. The quantum operators qˆn, fˆm are expressed in terms of creation and annihilation operators r ˆ ~ fn = aˆn† + aˆn , (4.19) 2ωnc and r ~ωnc qˆn = i aˆ† aˆn , (4.20) 2 n − lead to the quantization of the magnetic flux and charge density via Eq. (4.12) and (4.17). These operators obey the standard field commutation relation

h i X∞ qˆ(x), Φˆ (x0) = i~ φ(x)φ(x0) − n (4.21)

= i~δ(x x0). − − Ultimately, the quantum Hamiltonian for the transmission line resonator with ﬁnite length d is Z d 1 2 1 2 HTL = dx qˆ + (∂xφˆ ) , (4.22) 0 2c 2l and can therefore be simpliﬁed by expressing it in terms of the ladder operators, taking the familiar form,

X∞ 1 Hˆ = ~ω aˆ†aˆ + . (4.23) TL n 2 n=0

4.2 Dissipation from a 1D TransmissionLine Res-

onator

Consider a circuit quantum electrodynamic system composed of a superconducting transmission-line resonator coupled to a co-planar waveguide

65 which acts as an input/output channel. We aim to derive an expression for the quality factor of the transmission line resonator using a full quantum treatment of the coupled resonator-waveguide system. The classical

Figure 4.2: The resonator-waveguide system and its lumped circuit ana- logue. The resonator and waveguide can be considered as a sum of harmonic LC-circuits, where coupling capactiances at each end impose boundary conditions.

Lagrangian of this system can be written as a sum of the resonator and waveguide Lagrangians, with the addition of the capacitive coupling between them [64], i.e.,

= L + + , (4.24) LTot Res LWG LC where  2  n 1 1 X  φν φν 1  1 ˙ 2  ˙ 2 − −  ˙ 2 Res = Ccφ0 + crdzφν  + Ccφn, (4.25) L 2 2  − lrdz  2 ν=1  

 2  N 1 1 X  Φµ Φµ 1  1 ˙ 2  ˙ 2 − −  ˙ 2 WG = CcΦ0 + cwdzΦµ  + CcΦN, (4.26) L 2 2  − lwdz  2 µ=1  

66 and

= C φ˙ n Φ˙ . (4.27) LC − c · · 0

Here Cc is the coupling capacitance at each end of the resonator denoted by the subscript r, cr,w and lr,w are the capacitance and inductance per unit length of the resonator, cr and lr, and waveguide, cw and lw, respectively, and φn (Φn) is the magnetic flux at each point n in the respective lumped circuit element sequences associated with the resonator (waveguide), as indicated in Fig. 4.2. These expressions can be significantly simplified following diagonalization. An important approximation in the derivation of the corresponding Hamiltonian is the assumption that

C c 1. (4.28) cr,wLr,w

This criterion is generally fulﬁlled since the coupling capacitances must be small to allow for an appropriate build-up and storage of an electromagnetic ﬁeld inside the resonator. Writing the conjugate momenta ξn, Ξn of the variables qn, Qn leads to the Hamiltonian X X X X 1 h 2 2 2i 1 h 2 2 2i = q + ω ξ + Q + Ω Ξ + C ϕn(0)qn ψm(0)Qm. HTot 2 n n n 2 m m n c · n=0 m=0 n=0 m=0 (4.29) This expression describes two harmonic oscillators capacitively coupled at each end. Using the standard quantization procedure of a harmonic oscillator as shown in the previous section and by selectively exciting a single resonator mode, one obtains

X X ˆ = ~ωaˆ†aˆ + ~ Ωnbˆ†bˆn ~ gk bˆ† bˆk aˆ† aˆ , (4.30) H n − k − − n k=0

67 where the coupling strength gk can be expressed in terms of experimental parameters only, such that r r 1 1 1 1 p g = C ωΩ , k c c L q c L q k r r 1 + 2 Cc + C2Z2ω2 w w 1 + 2 Cc + C2Z2 Ω2 crLr c r cwLw c w k (4.31) where Zr,w is the impedence of the resonator and waveguide, respectively. It is known that the coupling strength should be proportional to the coupling capacitance and the impedance of the waveguide, since the influence of the transmission line on the resonator is generally due to its impedance [56]. In addition, the higher the energy of the mode excited/supported by the resonator and waveguide, the higher the strength with which they couple to each other. Assuming they are made from the same material, and that all terms C2 are negligible, the coupling constant ∝ c in Eq.(4.31) then simplifies significantly to

2 Cc gk Z0π √k, (4.32) ≈ ωr with the the characteristic impedence of the transmission line Z0. This approximation of the coupling strength gk has the correct dimension of a frequency, i.e., Hz, and is in accordance with what we would expect for the simpliﬁed case described here. Following the standard quantum Langevin approach for a harmonic os- ciallator coupled to a collection of electromagnetic reservior oscillators, i.e., the waveguide taken to be in thermal equilibrium, under the RWA, one arrives at a decay rate [39] Z X dn γ = 2π g2δ (ω Ω ) 2π g2δ (ω Ω ) . (4.33) n n ∆n n n n − → −

68 The frequency modes of the waveguide are deﬁned by a transcendental equation that follows from the wave-equation and boundary conditions of the Euler-Lagrange equations of motion. The solution of these equations can be obtained by a Taylor expansion up to ﬁrst order, under the previous assumption Eq.(4.28), such that

π n Ωn = · . (4.34) 2 Cc + 1 L √c l cwLw w w w If the waveguide and resonator are considered to be composed of the same material, cr = cw = c and lr = lw = l, and the waveguide is a semi-inﬁnite transmission line, L , for example, acting as a measurement channel, w → ∞ the resonator’s quality factor, Qext = ω/γ, can be written as

2 2 2 2 2 22 2Cc 1 + CcZ ω + cL 1 + CcZ ω Qext = 2 . (4.35) CcZω where Z = √l/c is the characteristic impedance. This quality factor is idealized and therefore tends to inﬁnity as the coupling capacitances approaches zero. However, if one wants to precisely match the true experimental parameters, an intrinsic saturation of the resonator’s quality factor towards which the idealized Q-factor tends [42] must be considered. A saturation occurs because of the presence of additional input (drive) and ouput (measurement) decay channels not considered in the theoretical treatment so far, which is phenomenologically accounted for via

1 1 1 = + , (4.36) Q Qint Qext by introducing an empirical quality factor saturation limit Qint.

69 (a) The calculated Q-Factor (b) The measured Q-Factor. From [42].

Figure 4.3: Comparison of the here calculated Q-factor and the measurement by Goeppl et al. Both curves show qualitatively similar behavior, whereas the precise values diﬀer because the details of the experimental parameters of the waveguide could only be estimated from the published information.

ω0 resonator fundamental angular frequency 14.7215 GHz

Lr length of the resonator 28.449 mm

10 c , c capacitance per unit length 1.27 10− F/m r w ·

7 l , l inductance per unit length 4.53 10− H/m r w ·

Q intrinsic quality factor 2.3 105 int ·

Table 4.1: The values used for the theoretical calculation of the Q-factor as published by Goeppl et al. [42].

Taking this into account and using the values listed in the table above for the waveguide and resonator parameters, the dependence of the resonator quality factor, Q, on the coupling capacitances, Cc, are obtained and dis-

70 played in Fig. 4.3 (a). This result is in excellent qualitative agreement with the experimental data and calculation by Goeppl et al. [42] as shown in Fig. 4.3 (b). The discrepencies between the calculation and experimental ﬁndings can be explained by the fact that it was necessary to estimate some of the parameters of the waveguide that were not explicitely stated in the article.

71 Chapter 5

Cooling a transmission-line resonator with a beam of Rydberg atoms

A hybrid quantum system composed of atoms prepared in circular Ryd- berg states coupled to a transmission line resonator oﬀers new opportunities for processing and storing quantum information. However, it also oﬀers possibilities for exploiting the Rydberg atoms or the resonator to engineer particular quantum states of interest of the other partner system. For example, the Rydberg atoms can be employed to add or remove photons from a complementary resonator mode on, or near, resonance with the Rydberg transition frequency in order to prepare particular Fock states of the resonator. Another possibility is to generate entanglement by exploiting a sequential interaction of multiple atoms with a selected resonator mode.

72 In this, and the following chapter, the ﬁrst of these schemes is studied. First by using an analytical method and then using a numerial approach in order to improve the scheme’s eﬃciency.

5.1 The atomic beam as an engineered quantum

reservoir

Recently, one particular focus of hybrid quantum system research has been the quantum state preparation of an individual system with the help of another quantum system [1]. To investigate the extent to which Rydberg atom beams can be employed to manipulate, and in particular cool, a selected mode of a transmission- line resonator, we now consider a beam of atoms prepared in circular Rydberg states crossing the resonator and thus interacting with its thermal electromagnetic field via their electric dipole transition moments. In this treatment we consider two selected circular Rydberg states with values of n differing by 1 in each atom. The lower selected state is considered as the ground state g , while the higher state is the excited state e . | i | i First we consider the internal dynamics of the atom-resonator system, in an analogous way implemented by Pielawa et al. [92]. Additionally, we include an open quantum system approach by accounting for thermal heating and dissipation of the resonator due to its coupling to a thermal bath. The problem will be studied on a coarse-grained time scale, ∆t, where ∆t is significantly longer than the average atom-resonator interaction time τ. If the arrival rate of atoms in state g in the resonator is defined as | i

73 r, then in order to ensure that at most one atom at a time interacts with the resonator, we must assume that rτ 1. The product r∆t 1 gives the average number of atoms ﬂying over the resonator during the time- interval studied. To characterize and analyze the dynamics of this system, we need to distinguish between two cases:

1) In the case of an atom being present above the resonator, the internal dynamics of the resonator-atom system is governed by the Jaynes- Cummings Hamiltonian [62] ˆ σˆ z ∆ = ~ωc + aˆ†aˆ + ~ σˆ z + ~g σˆ †aˆ + aˆ†σˆ , (5.1) H 2 2 | {z } | {z } ˆ ˆ 1 H0 H where the population diﬀerence operator is σˆ z = e e g g , and | ih | − | ih | σˆ g e and σˆ † e g are the pseudo-spin operators of the two- ≡ | ih | ≡ | ih | level atom – its lowering and raising operator, respectively. The

ﬁeld mode creation and annihilation operators are aˆ†, aˆ. The average interaction strength mediated by the electric dipole coupling is g, and

∆ = ω ωc describes the detuning between the transition frequency 0 − of the atom and the fundamental frequency of the resonator. We deﬁne the system and interaction Hamiltonian as above so that the commutator between them vanishes, i.e., h i ˆ , ˆ = 0. (5.2) H0 H1 Taking advantage of a vanishing commutator simpliﬁes the Baker-

Zˆ ˆ ˆ Campbell-Hausdorﬀ formula [24], e = eH0 eH1 , with

ˆ ˆ Zˆ = ln (eH0 eH1 ) (5.3) X∞ ˆ ˆ ˆ ˆ = + + Zm( , ), (5.4) H0 H1 H0 H1 m=2

74 ˆ ˆ where the Zm( , ) are homogeneous Lie polynomials, which are H0 H1 linear combinations of nested commutators of ˆ and ˆ , and equal H0 H1 zero in this case. This allows us to write the unitary time-evolution operator as

i ˆ t/~ Uˆ (t) = e− H

i ˆ t/~ i ˆ t/~ = e− H0 e− H1 . (5.5)

2) The second case of interest is that which arises if no atom is present above the resonator. In this situation the interaction between atom and transmission line resonator can be considered "switched oﬀ". Both the resonator and atom evolve independently according to the system Hamiltonian

Uˆ (t) = exp ( iHˆ t). (5.6) − 0

Taking both cases into account, we can quantify the evolution of the density matrix on a coarse-grained timescale, ∆t, without an atom present as

iHˆ 0(∆t) iHˆ 0(∆t) ρˆ0(t + ∆t) = e− ρˆ(t)e , (5.7) and with an atom interacting with the resonator for a time τ ∆t as

( iHˆ ∆t) h i (iHˆ ∆t) ρˆ (t + ∆t) = e − 0 exp ( iHˆ τ)ρˆ(t) exp (iHˆ τ) e 0 . (5.8) 1 − 1 1

Given the density matrix at time t, the total system evolves under the

Hamiltonian Hˆ 0 over a time step ∆t. The probability of this situation occuring is (1 r∆t), and corresponds to the situation in which there is no − atom interacting with the resonator. On the other hand, the probability r∆t

75 of an atom being above, and thus interacting with, the resonator marks the situation where the density matrix undergoes a change described by the

Hamiltonian Hˆ 1 during the interaction time τ. Therefore, transforming the problem into an interaction picture with respect to the system Hamiltonian

Hˆ 0, the change in the internal evolution of the density matrix due to the interaction with an atom during time step ∆t may be expressed as Z i ˆ τ/~ i ˆ τ/~ ρˆ (t + ∆t) = ρˆ (t) (1 r∆t) + r∆t dτP(τ) e− H1 ρˆ (t)e H1 , (5.9) tot tot · − tot where ρˆtot(t) is the evolution of the density matrix up to a time t, which represents the beginning of the time step to be evaluated . The probability distribution P(τ) in Eq. (5.9) represents the distribution of interaction times that arise because in a realistic experimental setting a typical ensemble of circular Rydberg atoms do not all move with the same velocity across the resonator. Therefore we have to integrate over all possible interaction times weighted by their probility of occurance, P(τ). Equ. (5.9) can be rearranged to yield an expression for the evolution of the total density matrix during each time interval ∆t Z ρˆtot(t + ∆t) ρˆtot(t) i ˆ τ/~ i ˆ τ/~ − = r∆t dτP(τ) e− H1 ρˆ (t)e H1 rρˆ (t). (5.10) ∆t tot − tot

This resembles the discrete form of the derivative of the reduced density matrix, giving its equation of motion on a coarse-grained time scale ∆t during which the interaction with a beam of atoms can occur. Since the atom is initially prepared in the ground state and both Hilbert-spaces are uncorrelated before the atom-resonator interaction, the initial density matrix can be factorized as a product of the reduced density matrix of the resonator, ρc(t), and the initial density matrix of the atom when entering

76 the resonator, which can be assumed to be in the ground state, ρtot(0) = ρ (0) g g . Provided no atom is present above the resonator, the total c ⊗ | ih | density matrix of the system evolves as ρ (t) = ρ (t) g g . tot c ⊗ | ih | However, if there is an atom present, the time-evolution operator takes on the form given in Eq. (5.5). Therefore we need to evaluate the eﬀect of the interaction Hamiltonian on the time-evolution of the atomic ground state g to obtain an expression for the evolution of total density matrix in terms | i of the reduced density matrix ρc(t) when an atom is present. In order to evaluate the time-evolution operator in the interaction picture, the series expansion of the exponential function needs to be applied,

k X∞ ( igHˆ 1τ) exp ( igHˆ τ) g = − g , (5.11) − 1 | i k! | i k=0 which requires the individual calculation of all higher-order terms of the ˆ n interaction Hamiltonian, H1 . The first six orders of the expansion are ∆ Hˆ = aˆσˆ † + aˆ†σˆ + σˆ , (5.12) 1 2g z !2 3 ∆ ∆ Hˆ = (aâˆ†)aˆσˆ † + (aˆ†aˆ)aˆ†σˆ + Hˆ + aâˆ† e e aˆ†aˆ g g , (5.13) 1 2g 1 2g | ih | − | ih | !4 !2 2 2 5 ∆ ∆ 3 Hˆ = aâˆ† aˆσˆ † + aˆ†aˆ aˆ†σˆ Hˆ + 2 Hˆ 1 − 2g 1 2g 1 ∆ 2 2 + aâˆ† e e aˆ†aˆ g g , (5.14) 2g | ih | − | ih | !2 2 ∆ Hˆ = aâˆ† e e + aˆ†aˆ g g + , (5.15) 1 | ih | | ih | 2g !2 !4 2 2 4 ∆ 2 ∆ Hˆ = aâˆ† e e + aˆ†aˆ g g + 2 Hˆ , (5.16) 1 | ih | | ih | 2g 1 − 2g !2 !4 !6 3 3 6 ∆ 4 ∆ 2 ∆ Hˆ = aâˆ† e e + aˆ†aˆ g g + 3 Hˆ 3 Hˆ + . 1 | ih | | ih | 2g 1 − 2g 1 2g (5.17)

77 For the expressions in Eq. (5.12) to (5.17) a re-occuring pattern for all the equations with even exponents can be identified, yielding a generating function in terms of a binomial series n ! !2k X n k n k 2n n ∆ − − Hˆ = aâˆ† e e + aˆ†aˆ g g . (5.18) 1 k 2g | ih | | ih | k=0 Using this expression, we can also determine a generating function for the interaction Hamiltonian with odd exponents via the matrix product ˆ 2n+1 ˆ 2n ˆ H1 = H1 H1, which leads to n ! !2k " ! !# X n k n k 2n+1 n ∆ − ∆ − ∆ Hˆ = aâˆ† aˆσˆ † + e e + aˆ†aˆ aˆ†σˆ g g . 1 k 2g 2g| ih | − 2g| ih | k=0 (5.19) To further simplify the expressions in Eq.(5.18) and (5.19), the binomial

n Pn n n k k expansion (x + y) = k=0 k x − y can be used, leading to,  !2n  !2n 2n  ∆   ∆  H1 = aˆaˆ† +  e e + aˆ†aˆ +  g g , (5.20)  2g  | ih |  2g  | ih |

 !2n !  !2n ! ˆ 2n+1  ∆  ∆  ∆  ∆ H1 = aˆaˆ† +  aˆσˆ † + e e + aˆ†aˆ +  aˆ†σˆ g g .  2g  2g| ih |  2g  − 2g| ih | (5.21) Ultimately, we obtain for the series expansion of the time-evolution operator

k X∞ ( igHˆ 1τ) exp ( igacHˆ τ) g = − g , (5.22) − 1 | i k! | i k=0  !2k X∞ 2k ( igτ)  ∆  = − aˆ†aˆ +  g (2k)!  2g  | i k=0  !2k X∞ 2k+1 ( igτ)  ∆  ∆ − aˆ†aˆ +  g − (2k + 1)!  2g  2g| i k=0  !2k X∞ 2k+1 ( igτ)  ∆  + − aˆaˆ† +  aˆ e . (5.23) (2k + 1)!  2g  | i k=0

78 n n P ( 1) 2n+1 P ( 1) 2n − − Using the series expansion for sin x = (2n+1)! x and cos x = (2n)! x , this can be further simplified to  s  !2   ˆ  ∆  exp ( igH1τ) g = cos gτ aˆ†aˆ +  g − | i  2g | i ! q 2 ∆ sin gτ aˆ†aˆ + ∆ 2g i g q 2 − 2g ∆ | i aˆ†aˆ + 2g ! q 2 ∆ sin gτ aâˆ† + 2g i aˆ e . (5.24) q 2 − ∆ | i aâˆ† + 2g After taking the effect of the interaction Hamiltonian on the factorized total density matrix into account, we calculate the partial trace of the total density matrix with respect to the atomic degrees of freedom ρˆc (t) = Tra ρˆtot (t) . In the limit of small discrete time steps, this yields a differential equation, ∂ρˆc/∂t, from Eq. (5.10). In addition to the resonant/near-resonant interaction with the atoms, the transmission-line resonator is also subject to dissipation. However, because of the small decay rate of the resonator with typical Q-factors of 106 as has been shown in Chapter 4, this can be neglected if a compar- ∼ itively short interaction time between a single atom and the resonator is considered. In this limit we can linearly add [20] the contribution of the resonator-dissipation due to its coupling to a thermal bath with temperature T, γ Dth = [n¯ r(ωc, T) + 1] 2aˆρˆcaˆ† aˆ†aˆρˆc ρˆcaˆ†aˆ 2 − − (5.25) γ + n¯ (ω , T) 2aˆ†ρˆ aˆ aâˆ†ρˆ ρˆ aâˆ† , 2 r c c − c − c 79 where γ is the photon dissipation rate of the resonator, to the equation of motion of the internal dynamics. This yields the master equation of the reduced density matrix of the resonator coupled to a beam of Rydberg atoms initially prepared in the circular ground state g , and a thermal bath, | i which obtained to be

Z q q dρˆc τ τ =r dτP(τ) cos 4g2aˆ aˆ + ∆2 ρˆ (t) cos 4g2aˆ aˆ + ∆2 dt 2 † c 2 † τ p 2 2 τ p 2 2 sin 4g aˆ†aˆ + ∆ sin 4g aˆ†aˆ + ∆ 2 2 2 + ∆ ρˆc(t) p 2 2 p 2 2 4g aˆ†aˆ + ∆ 4g aˆ†aˆ + ∆

p τ 2 2 q sin 2 4g aˆ†aˆ + ∆ τ 2 2 + i∆ ρˆc(t) cos 4g aˆ†aˆ + ∆ p 2 2 4g aˆ†aˆ + ∆ 2

p q τ 2 2 sin 2 4g aˆ†aˆ + ∆ τ 2 2 i∆ cos 4g aˆ†aˆ + ∆ ρˆc(t) p 2 2 − 2 4g aˆ†aˆ + ∆ τ p 2 2 τ p 2 2 sin 4g aâˆ† + ∆ sin 4g aâˆ† + ∆ 2 2 2 + 4g aˆρˆc(t)aˆ† rρˆc(t) p 2 2 p 2 2 4g aâˆ† + ∆ 4g aâˆ† + ∆ −

γ + [n¯ (ω , T) + 1] 2aˆρˆ aˆ† aˆ†aˆρˆ ρˆ aˆ†aˆ 2 r c c − c − c

γ + n¯ (ω , T) 2aˆ†ρˆ aˆ aˆaˆ†ρˆ ρˆ aˆaˆ† . 2 r c c − c − c (5.26)

The probability distribution P(τ) represents the disitribution of interaction times that arise because the atoms do not all move with the same velocity across the resonator. Therefore we have to integrate over all possible interaction times weighted by their probility of occurance, P(τ).

80 While the first two terms in the integral kernel describe the excitation of the resonator state due to the interaction with the atomic beam, the third and fourth part show the effect of the accumulation of phase differences between each resonator state resulting from this interaction. The fifth, and last, term describes the coherent decay of the resonator state n + 1 | i to n as the atoms in their ground state are excited and subsequently | i carry away the excitation. This coherent atom-resonator interaction can be exploited to remove photons from a selected resonator mode, effectively cooling the resonator. The factors in the last two lines in Eq. (5.26) are the Lindblad operators accounting for the thermal decay and heating processes affecting the resonator because of its coupling to the environment, e.g. a waveguide to connect the resonator to a measurement device, as described in Chapter 4.

5.2 Resonator De-Excitation

In the hybrid quantum electrodynamics experiments with gas-phase Ry- dberg atoms and microwave circuits pertaining to the work here, it is of interest to de-populate a selected resonator mode and consequently lower the resonator’s eﬀective temperature. The goal of such a cooling process is to establish a non-equilibrium steady-state between the atomic beam carrying away excitations from the resonator and the heating due to the coupling to the local environment via a waveguide. This method of resonator cooling is expected to be achievable experimentally because many of the individual components necessary to realize this proposed scheme are already in place [66, 67, 51].

81 To investigate this approach to resonator cooling it is necessary to characterize the probability of the population of each Fock state in the resonator, p(n). This is done by considering the diagonal elements of the reduced density matrix in the photon number basis, p(n) = ρnn = n ρˆ n . For this h | | i treatment, and without loss if generality, we assume that interaction times of the individual atoms in the beam with the resonator mode of interest follow a Gaussian distribution, P(τ), centered on an average interaction time τ¯, such that (τ τ¯)2 1 − P(τ) = e− 2σ2 . (5.27) √2πσ2 The time-evolution of these probabilities in the combined atom-resonator system is described by a set of coupled diﬀerential equations of the form

p˙(n) = Anp(n) + Bn+1p(n + 1) + Cn 1p(n 1), (5.28) − − where the prefactors are evaluated [17] to be

1 2 2 2 σ2 4g2n+∆2 p 2 2 2g n + ∆ + 2g n e− 2 ( ) cos τ¯ 4g n + ∆ An = r 1 γ (2n¯ T + 1) n + n¯ T , 4g2n + ∆2 − − | {z } | {z } An(r) An(γ) (5.29)

1 2 σ2 ∆2+4g2n p 2 2 2g n 1 e− 2 ( ) cos τ¯ 4g n + ∆ B = r − + γ (n¯ + 1) n, (5.30) n 4g2n + ∆2 T

Cn = γn¯ T(n + 1). (5.31)

These prefactors can be interpreted as rates describing the transitions between diﬀerent Fock states of the resonator. As depicted in Fig. 5.1, Bn corresponds to the rate of depopulation of the selected resonator mode because of the interaction with the atomic beam, i.e., Bn r, and the decay ∝

82 Figure 5.1: The population transfer rates An, Bn and Cn among three

consecutive Fock states in the resonator. It can be seen that Bn corresponds to a de-excitation of the resonator from the Fock state n to the Fock state | i n 1 , whereas Cn describes an excitation from the state n to the state | − i | i n + 1 . The total rate of de-population of the Fock state n is therefore | i | i An = Bn + Cn. Together with the competing transfer rates Bn+1 and Cn 1 − populating the n Fock state, the rate of change of the population of that | i state n can be calculated according to Eq. (5.28). | i of the resonator photons into the waveguide, i.e., Bn γ. The eﬀective res- ∝ onator cooling rate results from the competition between Bn and Cn, which represents the excitation of the resonator by thermal photons entering it from the waveguide. The set of coupled diﬀerential equations described in Eq.(5.28) can be written in matrix form as

p˙ (t) = Mp(t), (5.32)

83 where p(t) is a vector, the elements of which correspond to the occupation probability of the respective Fock state, and M is the corresponding matrix containing the transfer rates, i.e.,          p (t)  A B   0   0 1                   p1(t)  C0 A1 B2                   p2(t)   C1 A2 B3 0                   p3(t)   C2 A3 B4      p(t) =   , M =   , (5.33)      .   .   .   ..   .                    pn 2(t)  0 Cn 3 An 2 Bn 1   −   − − −              pn 1(t)  Cn 2 An 1 Bn   −   − −               pn(t)   Cn 1 An − with the general solution

Mt p(t) = pss + e p(0) pss . (5.34) −

This depends on the non-equilibrium steady state solution pss, which is the state the system ends up in as t , and on the initial state of the → ∞ resonator p(0), which is generally assumed to be the thermal state of the system under consideration being in thermal equilibrium with the local environment.

84 5.2.1 Steady State Solution

We ﬁrst solve for the steady state, i.e., the resonator mode of interest in dynamic equilibrium resulting from the competing processes of the introduction of photons from the thermal bath, and excitations being carried away by the atoms. The probability of zero photons in the resonator can be obtained analytically p0 = n 2 2 X∞ Y γn¯ T 4g x + ∆ 1 2 σ2(∆2+4g2x) p 2 2 2 2 2g r 1 e− 2 cos τ¯ 4g x + ∆ + γ (n¯ T + 1) 4g x + ∆ n=1 x=1 − 1 − + 1 .

(5.35)

The steady state photon number distribution in the transmission line resonator can be calculated via the recursive relation n Y Cj 1 − pn = p0 . (5.36) Bj j=1 Considering the n = 70 n = 71 transition between the circular | i ↔ | i Rydberg states, which occurs at a transition frequency of ν = 18.78 GHz, a resonator coupling strength of g/(2π) = 4.25 MHz is assumed. This is experimentally achievable for Rydberg atoms positioned at a distance of 10 µm above the center conductor [80]. The assumed resonator’s quality factor of Q 106 leads to a dissipation ≤ rate of γ 20 kHz. Note that currently experimentally achievable quality ≥ factors for the here given initial temperature T = 4 K of the resonator are on the order of Q . 105 [8] because the condition for the resonator material to be fully superconducting, Tc T, is not fulﬁlled at this temperature. 85 ●

0.8

0.6

p ( n ) 0.4

0.2

● 0.0 ● ● ● 0 1 2 3 4 Photon Number n

Figure 5.2: Distribution of the photon occupation number for a fundamental frequency of ν = 18.78 GHz. The red dots mark the occupation probability of the Fock state n of the resonator in the non-equilibrium steady | i state. The Planck distribution for an effective temperature Teff = 360 mK is plotted (blue line), showing the qualitative fit with the steady state photon distribution.

Assuming an average interaction time of the atoms with the resonator of τ¯ = 50ns, there are two distinct cases to be considered. First a coherent beam of atoms is considered, where the atomic velocities are all equal, and therefore their interaction time distribution can be modelled by a dirac delta function P(τ) = δ(τ τ¯) with −   + , τ = τ¯  ∞ δ(τ τ¯) =  (5.37) −  0, τ , τ.¯

86 1.0 No Atom Beam 0.8 Coherent Beam

0.6 p ( n ) 0.4

0.2

0.0 0 1 2 3 4 5 6 Photon Number n

Figure 5.3: Depletion of the fundamental mode of the resonator. The thinner red bars represent the occupation probability distribution of the Fock states n at T = 4 K. This results in an average photon occupation | i number of n = 3.96. The thicker blue bars represent the photon num- h ith ber distribution after the interaction with an atomic beam with a ﬁxed atom-resonator interaction time τ¯, leading to a thermal photon number

distribution with an average number of photons n ss = 0.122. h i

For the case in which the atomic transition frequency is resonant with the fundamental mode of the resonator, i.e. ∆ = 0, we calculate the photon number probability distribution in the nonequilibrium steady state. Assuming an atomic rate of r = 106 atoms per second, the Fock state occupation probability follows a thermal photon number disitribution for this selected mode, as can be seen in Fig. 5.2. Therefore, a ﬁtted parameter

87 representing the effective temperature of the resonator Teff [93] 1 n n p (n) = h ith , (5.38) th n + 1 n + 1 h ith h ith where 1 n th = . (5.39) h i exp ( hν ) 1 kBTeff − is introduced. From this calculation, it can be seen in Fig. 5.3 that the average number of photons, and thus the effective temperature, of a selected resonator mode can be reduced significantly by coupling the resonator to a continuous beam of flying atoms. The initial state of the resonator when cooled cryogenically using liquid helium may be described by the thermal photon number distribution at T = 4 K, as indicated by the red bars in Fig. 5.3. The average number of photons contained in the resonator is therefore n = 3.96. After allowing atoms, prepared in the lower circular h ith 6 1 Rydberg state, to cross the resonator at a rate r = 10 s− , this average photon number can be reduced since the atoms will continuously carry away excitations. Assuming a coherent atomic beam with a constant interaction time τ, the average number of photons in the non-equilibrium steady state can be reduced to n = 0.122, corresponding to an effective temperature h iss of the resonator of Teff = 0.36 K. This method of cooling a selected resonator mode depends on several experimental parameters that affect its efficiency. These are studied here to identify optimal experimental conditions under which to implement this cooling process. In doing this, the fidelity of the vacuum state of the resonator, = 0 ρˆ 0 , is calculated. This fidelity is F0 h | | i assumed to be a valid measure of the resonator de-excitation. First, the dependence of the ground state fidelity on the detuning of the atomic transition frequency from the frequency of the fundamental mode

88 0.9

Incoherent Beam

0 0.8 Coherent Beam 0.7

0.6

0.5

0.4

Groundstate Fidelity ℱ 0.3

0.2 -1.0 -0.5 0.0 0.5 1.0 DetuningΔ(GHz)

Figure 5.4: The dependence of the 0 Fock state fidelity on the detuning | i of the atomic transition frequency and the resonator resonance frequency of the selected fundamental mode. of the resonator is investigated. The observed behavior, which can be seen in Fig. 5.4, is strongly dependent on the velocity distribution of the atomic beam. As can be seen in the figure, for the case of a coherent beam of atoms that has been considered up to now, for which the interaction time of the atoms with the resonator is assumed to be constant, the dependence of the ground state fidelity on the detuning is described by the Fourier transform of a square signal pulse. The alternative case, also considered in Fig. 5.4, is that of an incoherent beam of atoms, in which the interaction times follow a normal distribution with a broad width, chosen here to be σ = τ/¯ 3. Since 99.7% of all values lie within three times the standard deviation, this assumption ensures that only 0.15% of all possible values for the individual interaction time

89 0.9

0.8 0

0.7

0.6

0.5

0.4 g = 62.82 MHz Groundstate Fidelity ℱ 0.3

0.2 0 20 40 60 80 100 120 140 Interaction Strength g(MHz)

Figure 5.5: The dependence of the resonator ground state ﬁdelity on the atom-resonator coupling strength. A coherent beam (solid blue line) reaches local minima for speciﬁc values of the interaction strength, one being at g = 62.82 MHz. The minima represent trapping states. The incoherent atomic beam (dashed red line) does not exhibit such behavior,

with a roughly constant fidelity of 0 0.85 for values of g 80 MHz. F ≈ ≥ of each atom with the resonator are mathematically τ 0, and hence not ≤ physical, justifying the assumption of such a probability distribution. The corresponding calculation for this case shows that the dependence of the ground state fidelity on the atom-resonator detuning follows a Lorentzian function. This is the Fourier transform of an exponential signal. One can therefore postulate that measuring the dependence of the ground state fidelity on the detuning can, for example, provide information on the interactions between the atoms and the resonator and thus about the velocity distribution of a given beam of atoms.

90 Second, the ground state occupation probability of the resonator is investigated as the coupling strength between the Rydberg atoms and the microwave resonator is adjusted. In this case the observed behavior shows a fundamental difference between a coherent and incoherent beam of atoms. The former exhibits the existence of trapping states [77], because the constant interaction time leads to situations in which the average photon number in the resonator approximately fulfills the condition for a full Rabi cycle for all atoms, p gτ n 2π k, (5.40) h i ≈ · where k N. This renders the preparation of the vacuum state of the ∈ resonator highly inefficient because ground state atoms entering the resonator will leave the resonator again when being close to the ground state, and thus do not carry away any significant amount of resonator excitations. The photon number distribution of this anomalous steady state for an atom-resonator coupling value of g = 62.82 MHz is compared to that for a thermal state with the same average photon number n in Fig. 5.6. The h i non-equilibrium steady state achieved in this way is almost entirely an even statistical mixture between the 0 and 1 Fock states, where p(n = 0) 0.5 | i | i ≈ and p(n = 1) 0.4, with the occupation probability of other states being ≈ P n=2 p(n) . 10%. This shows the possibility of controlled state preparation of the resonator by choosing appropriate experimental parameters. In the latter case of an incoherent atomic beam, however, such trapping states are not observed. Due to the wide spread, σ = τ/¯ 3, in interaction times, instances for which the average number of photons enhancing the interaction strength leads to a full Rabi oscillation do not occur for a consid-

91 0.6 Steady State 0.5 Thermal State

0.4

p ( n ) 0.3

0.2

0.1

0.0 0 1 2 3 4 5 6 Photon Number n

Figure 5.6: Photon number distribution corresponding to the annomalous trapping state that arrises for an atom-resonator coupling strength of g = 62.82 MHz. erable amount of atoms whose interaction times differ significantly from the mean value τ¯ = 50 ns. The interactions of these atoms eventually carry away excitations leading to a dephasing of these trapping states. It should be mentioned here that the observation of such trapping states would re- quire experimtal developments beyond those currently realized for hybrid cavity QED with circular Rydberg states coupling to transmission line resonators. As previously mentioned, another important experimental factor affecting the efficiency of the de-excitation of the resonator is its quality factor. The dependence of the average number of photons in the non-equilibrium steady state is plotted in Fig. 5.7. It illustrates the high sensitivity of the average steady state photon number on the change in quality factors between

92 4 τ = 10ns r=5⨯10 6 ss

3 τ = 50ns r= 10 6

1 Average Photon Number 〈 n 〉

0 3 4 5 6 7

Quality Factor log10(Q)

Figure 5.7: The dependence of the steady state average photon num-

ber , n ss, on the resonator quality factor, Q. The product rτ 1 ensures h i the approximation of at most one atom interacting with the resonator is fulﬁlled.

Q = 104 and Q = 106. Significant de-excitation to a steady state average photon number of n 0.5 can be achieved with the here proposed h iss ≤ method for a quality factor of Q 5 105. ≥ · The final state of the resonator after the interaction with the atomic beam is a non-equilibrium steady state which is determined by the competing processes between the thermal heating of the resonator, which is caused by the introduction of photons from the thermal bath at T = 4K, and the rate at which atoms carry away excitations. This leads to a detailed balance for a specific average photon number n in the resonator. Therefore, the rate at h i which atoms cross the resonator, r, is the main experimentally controllable

93 parameter that has to be studied in order to optimize the efficiency of populating the vaccuum state of the resonator. Phenomenologically, the more excitations that can be carried away before a thermal photon re-enters into the resonator due to its coupling to the environment, the lower the final effective temperature that can be achieved. Fig. 5.8 shows the exponential relationship between the average photon number in the steady state, n ss, h i and the rate at which the atoms cross the resonator. From this figure it can be seen that the higher the atomic beam rate, the more efficient the cooling process. This behavior is qualitatively similar for different average interaction times τ¯. However, the region in Fig. 5.8 in which the results are indicated by the dashed curves are beyond the limits of the approximation made in the calculation. These graphs therefore only give an indication of the powerful effective cooling that may be studied if the calculations are extended beyond the one-atom approximation made in this chapter. If the atomic beam is replaced with a dense pulsed beam for which the calcuations need to allow for more than one atom interacting with the resonator at the same time, we expect that it will be possible to prepare the resonator at much lower effective temperatures which will persist for a sufficient period of time to allow strong coupling experiments with much less than 1 thermal photon present in the resonator. This is the case that will be considered numerically in Chapter 6.

5.2.2 Time-Resolved Solution

In order to obtain a realistic estimate of how long it takes for a system being cooled by an atomic beam to reach its steady state using this approach,

94 4

ss τ = 10ns

3 τ = 50ns

1 Average Photon Number 〈 n 〉

0 0 2 4 6 8 10 Atom Beam Rate log(r)

Figure 5.8: Dependence of the average photon number n of the nonequi- h i librium steady state of the resonator on the rate of atoms crossing the resonator with diﬀerent average interaction times τ¯ as indicated. The continuous curves represent the values for r that fulﬁll the approximation rτ 1, where the probability of having two or more atoms interact with the resonator at the same is less than 0.25%. The dashed curves represent the region in which this approximation breaks down because the probability that two or more atoms interact with the resonator at the same time is not negligible anymore. we obtain a time-dependent solution of Eq. (5.34). The matrix M in this equation is a tridiagonal matrix. A useful property of such matrices is that

Mn n Mn n = Bn Cn 0, (5.41) , +1 +1, +1 ≥

95 which guarantees their eigenvalues to be real. The elements of M in Eq.(5.33) can be expressed as

Mn,m = δn,mAn + δn,m+1Cn 1 + δn,m 1Bn+1. (5.42) − −

Since the steady-state of a system is deﬁned as being unchanged under the evolution over time, the application of the matrix generating the dynamics of the system in the steady-state is Mpss = 0, which we can use to obtain

p˙ (t) = Mp(t) (5.43) = Mp(t) Mpss, − or equivalently p˙ (t) = M p(t) pss . (5.44) − This leads to the solution

Mt p(t) = pss + e p(0) pss , (5.45) −

X P 1 k ∞ where the matrix exponential must be calculated using e = k=0 k! X . This calulation can be carried out by diagonalizing the matrix M according to

1 D M 1 D = UMU− e = Ue U− , where U is a block matrix containing the → eigenvectors of M, allowing for a unitary transformation of the matrix, and D is a diagonal matrix containing the corresponding (negative) eigenvalues. It can be shown that the solution obtained in this way is valid, since the derivative of Eq. (5.34) is

Mt p˙ (t) = M e p(0) pss + pss Mpss − − | {z } |{z} =p(t) =0 (5.46)

= Mp(t),

96 and therefore a solution to Eq. (6.1).

The reason to incorporate pss into this solution, even though in the problem it simply means adding a zero vector, 0, is that it is necessary to ensure that a non-trivial steady-state solution exists. Without including the steady-state vector, any solution generated is going to produce a trivial steady-state solution pss = 0. However, because all the eigenvalues of M are real and negative, as shown above, a unique and stable solution of the system’s state must exist [105]. Hence, by ﬁnding any solution by solving the equation of motion, given its initial state at t = 0 and resproducing its steady-state for t , we have found the unique solution to the problem. → ∞ Since we assume that the initial state p(0) respresents the thermal distribution of the resonator mode at time t = 0, and the steady-state solution pss is readily calculated as seen in the previous section, the full solution of p(t) at all times t can be obtained. As can be seen from the previous investigation of the experimental parameters, the most eﬀective cooling is achieved for zero detuning, i.e., ∆ = 0, and a coherent beam of atoms, so that

r An = cos 2gτ¯ √n 1 γ (2n¯ T + 1) n + n¯ T , (5.47) 2 − − r h i Bn = cos 2gτ¯ √n 1 + γ (n¯ T + 1) n, (5.48) −2 −

Cn = γn¯ T(n + 1). (5.49)

The time evolution of the average photon number of the resonator is plotted in Fig. 5.9. The non-equilibrium steady state with n = 0.122 is h iss reached within approximately 100 µs. After having cooled the system close to its steady-state, it is also of interest

97 4.0

n 3.5

r e 3.0 b m

u 2.5 N

o 2.0 t o h

P 1.5

e g

a 1.0 r e v

A 0.5

0.0 0.01 0.1 1 10 100 Time t ( s)

Figure 5.9: Time-dependent evolution of the average photon number n of the resonator. The initial state of the resonator with a fundamental h i frequency of ν = 18.78 MHz is assumed to be a thermal state at Tenv = 4 K.

6 1 The atoms on resonance, ∆ = 0, cross the resonator at a rate r = 10 s− and couple to it for a constant interaction time τ¯ = 50 ns with a coupling strength g = 26.66 MHz. to know how quickly the system heats up again after the atomic beam is switched out of resonance, which is accounted for by changing the parameter r = 0. This is the time during which, in principle, experiments with an initially empty resonator could be performed. In Fig. 5.10 it can be seen that for a time t . 5µs the resonator contains less than n = 0.5 h i photons. This is suﬃcient to perform gate operations within the context of this experimental setup.

98 1.0 n

e 0.75 b m u N

o 0.5 t o h P

e g

a 0.25 r e v A

0.0 0.01 0.05 0.1 0.5 1 5 10 Time t ( s)

Figure 5.10: Time resolved reheating of the resonator.

It is important to note that the result arrived at in this chapter requires restricting the calculations to at most one atom interacting with the resonator at any time. This is a signiﬁcant limitation of the cooling process. Once the atomic rate r grows so high that the probability of two or more atoms interacting with the resonator at the same time cannot be neglected, our calculation must be modiﬁed.

99 Chapter 6

Atomic Beam Interaction: Numerical Approach

The analytical approach to investigating the cooling of selected resonator modes using beams of Rydberg atoms presented in Chapter 5 provides and accurate description of the dynamics of the system when there is at most one atom interacting with the resonator. However, this restriction on the number of interacting atoms places a limit on the rate, r, at which atoms can cross the resonator and in turn limits the eﬀective cooling of the microwave resonator’s fundamental frequency mode. To extend the analysis of the proposed hybrid system beyond this limit, we develop a numerical approach in this chapter that allows several atoms to interact with the microwave resonator at the same time, as illustrated in Fig. 6.1, permitting access to a parameter range that leads to much more eﬃcient cooling of the resonator mode.

100 g r

thermal bath

Figure 6.1: A schematic diagram of the experimental setup, allowing for more than one atom to interact with the microwave resonator at any given time t.

6.1 Modeling the Beam of Atoms

In order to numerically model a realistic experimental setting, it is necessary to account for random arrival times of atoms at the microwave resonator. This in turn leads to a random number of atoms interacting with the microwave resonator at any given time. To study the time-evolution in this situation, the dynamics are investigated in very small time steps. After each such time step, ∆t, the probabilty of a new number n atoms arriving at the resonator is calculated, as well as the probabilty of any atom that

101 has been interacting with the resonator during that time step to leave the resonator. A sketch of the algorithm created to model such behavior is shown in Fig. 6.2. The probability of one atom interacting with the resonator within each time step can be expressed as

P(∆t) = r ∆t, (6.1) · where r is the rate of atoms crossing the resonator per unit time, and ∆t is the time step size. In principle this probability P(∆t) can be inﬁnitely large, as the time step is arbitrarily chosen. However, over the duration T of an experiment, which covers N time steps, this eﬀect is canceled out if the rate r is uniformly distributed, and therefore constant, during the experiment. This can be seen shown by

Z T N¯ atoms = r dt 0 Z T/N Z 2T/N Z T = r dt + r dt + ... + r dt 0 T/N (N 1)T/N − Z T/N (6.2) = N r dt 0 | {z } =P(∆t)

= r T, · where N time steps occur with intervals ∆t = T/N. The probability of an atom interacting with the resonator during each time step is given in Eq. (6.1). Following from this deﬁnition of the probability of a single atom arriving at the resonator, we can extend this to the probability of two, three

102 t = Δt

0 Atoms

0 Atoms 1 Atom 2 Atoms 3 Atoms

t = 2Δt 0 1 0 1 2 0 1 2 3

0 Atoms 1 Atom 2 Atoms 3 Atoms

t = 23Δt 0 1 0 1 2 0 1 2 3

0 Atoms 1 Atom 2 Atoms 3 Atoms

t = NΔt = T ...

Figure 6.2: The ﬂow chart of the numerical approach. Within each time step ∆t the number of atoms leaving the microwave resonator is calculated

(orange arrows) after having interacted over a time tint. This depends on how many atoms remain within the resonator, and how many new atoms enter the microwave resonator (black arrows), based on their arrival prob-

ability Pn(∆t). The black arrows from subsequent orange atom number boxes, have been omitted for clarity of the sketch. or even n atoms arriving at the same time as

n Pn(∆t) = (r ∆t) , n 1. (6.3) · ≥ Therefore we need to estimate the error resulting from the need to introduce a limit on the maximum number of atoms considered to simultaneously interact with the microwave resonator at any given time step to ensure

103 computational feasibility. To obtain an order of magnitude estimate of the error, we compare the exact probability of zero atoms arriving at the resonator P0(∆t) within a given time step to the probability based upon the above approximation.

1. Exact:

X∞ P (∆t) = 1 (r∆t)k 0 − k=1 (6.4) 1 2r∆t = − . 1 r∆t − 2. Truncation to N atoms: XN P (∆t) 1 (r∆t)k 0 ≈ − k=1 (6.5) 1 2r∆t = − + . 1 r∆t − The sums in Eq.(6.4) and (6.5) are solved as converging geometric series for r∆t < 1. The error resulting from the truncation that arises from the consideration of at most N atoms interacting with the microwave resonator at any given time can be quantiﬁed as

(r∆t)N . (6.6) ≈ 1 r∆t − The larger the atomic rate r and/or the time step ∆t, the bigger the error will be. On the other hand, the more atoms N that are considered, the more exact the simulation is. For our purposes, an atomic rate of r = 20, a time step size of ∆t = 0.01 and a truncation to N = 5 atoms leads to an error of 0.04% in reproducing the probability of having no atom arrive at the ≈ resonator within a given time step. This low estimate of the error therefore justiﬁes the validity of this approach. A more detailed justiﬁcation

104 of this approximation by taking the mean interaction time τ into account and calculating the probabilty of having N 5 atoms interacting with the ≤ resonator at any given time can be found in Appendix A.

The probability of an atom exiting the resonator, Pex(τ), is cumulatively Gaussian distributed, whose mean is given by the average interaction time, τ¯, and therefore calculated based on the time it has so far been interacting with the resonator,

Z τ 1 (t τ¯)2 − 2 Pex(τ) = e− 2σ dt. (6.7) √2πσ2 −∞

In the case of a coherent beam of atoms, meaning a constant interaction time for each individual atom, and therefore σ = 0, the distribution of the interaction time can be modelled by a dirac delta function, which leads to the probability of an atom exiting the resonator to be

Z τ P (τ) = δ(t τ¯) dt ex −  −∞  0, τ < τ¯  =  . (6.8)  1, τ > τ¯

The description of the atom beam statistics in terms of a random distribution is based on Monte Carlo simulations, where one numerically simulates a single random realization, a trajectory, and then in turn averages over a large number of trajectories to get the expectation values of the random processes. This approach intrinsically accounts for the Gaussian distribution of the atoms in the beam and consequently their arrival times at the microwave resonator, as can be seen in Fig. 6.3.

105 = 85.47 = 8.64 0.04

0.03

0.02 Normalized Count

0.01

0.00 60 70 80 90 100 110 Number of Atoms

Figure 6.3: The probability distribution representing the number of atoms interacting with a resonator over a time T = 2 µs, with an average interaction time of τ¯ = 10 ns and a rate of r = 40 atoms/µs. The simulation was carried out for 500 trajectories.

6.2 Stochastic Dynamics

The mathematical framework for this approach is based on the theory of quantum dynamical semigroups [3]. We aim to ﬁnd a formal solution to the quantum master equation

∂ ρ(t) = ρ(t), (6.9) ∂t L a Markov semigroup which is a completely positive dynamical map φt that translates an initial density matrix onto a later density matrix within the same Hilbert space with a time parameter t 0. ≥

106 6.2.1 The stochastic process as a series of time steps

Since the Lindblad master equation is a linear superoperator, deﬁned as the time derivative of a density matrix,

φ∆t(ρ) φ0(ρ) [ρ] = lim − , (6.10) ∆t 0 ∆t L → one can use its property

φs(φt(ρ)) = φt+s(ρ), (6.11) to arrive at a mapping with the formal solution

s φt+s(ρ) = eL φt(ρ). (6.12)

This means going from one point in time t0, to a later point t1 = t0 + ∆t, the formal solution for the dynamics over the small time step ∆t is

0∆t ρ(t1) = eL ρ(t0), (6.13) which one can in turn take as the initial density matrix to get the formal solution after another time step ∆t.

1∆t ρ(t2) = eL ρ(t1) 1∆t 0∆t = eL eL ρ(t0) . (6.14)

This method can be applied analogously to any number of time steps until tN = N ∆t = T if we deﬁne t = 0. Thus, this constitutes a Markov chain · 0 of N steps, where given a solution to the density matrix at any time t0 we can extend the solution to any later time t, by solving the dynamics self-consistently for each individual time step depending on the solution of the quantum master equation during the previous time step.

107 6.2.2 The dynamics for a constant number of atoms

The dynamics during a single time step is governed by the quantum master equation in the interaction picture,

∂ρˆ h i √γc ˆ (N) =i ρ,ˆ Hi + (n¯ + 1) 2aˆρâˆ† aˆ†aˆρˆ ρâˆ†aˆ ∂t 2 − − (6.15) √γc + n¯ 2aˆ†ρâˆ aâˆ†ρˆ ρââˆ† , 2 − − where the interaction Hamiltonian,

XN ˆ (N) (n) (n) Hi = g aˆ†σˆ + aˆσˆ + , (6.16) − n depends on the number of atoms N interacting with the microwave resonator over that time step ∆t. Here we have assumed that over the small interaction time τ 1/γa the atoms do not decay, therefore we only consider the dissipation of the microwave resonator. By deﬁning the density matrix in vector form1, ρ~, the quantum master equation (6.9) with the Liouvillian super operator can be rewritten as L XN ∂ 1 (n) 1 (n) (n) 1 (n) 1 ρ~ = ig aˆ†σˆ + aˆσˆ + aˆ†σˆ aˆσˆ + ρ~ ∂t − − n ⊗ ⊗ − ⊗ − ⊗ · 1 1 (6.17) +γc(n¯ + 1) a a† a†a 1 1 a†a ρ~ ⊗ − 2 ⊗ − 2 ⊗ · 1 1 +γcn¯ a† a aa† 1 1 aa† ρ,~ ⊗ − 2 ⊗ − 2 ⊗ · where the product between the atomic and the photonic operators is a tensor product of each of their Hilbert spaces, and the photonic operator alone is a tensor product with the identity of (2N 2N)-dimensions. This × 1which can be done computationally by the ﬂatten command in several programming languages, including Python’s numpy module.

108 equation of motion for a constant number of atoms N can be formally solved for an initial time evolution t = 0 ∆t as 7→

∆t ρ~(∆t) = eL ρ~(0), (6.18) where ρ(0) is assumed to be the thermal state of the system at a temperature of the coupled bath. By truncating the bosonic Hilbert space of the photon operator a in matrix form, which is reasonable for a ﬁnite temperature of the bath where the higher dimension Fock states’ occupation probability is negligible, the solution for the density matrix ρ~(∆t) becomes analytically and computationally feasible. Therefore we solve for the dynamics of the microwave resonator over the interaction time τ numerically, integrating the solution over a single time step ∆t τ, and use the result as the initial density matrix to solve for the subsequent time step. With this method the dynamics during an atom-photon interaction for a number of atoms, n, prepared in their respective ground states, is obtained.

6.2.3 A beam of atoms interacting with the resonator

It is important to note that once an atom leaves the resonator, we no longer consider it part of the system, having carried away its excitation without being measured. This means that any correlations between the system and the departed atom that have built up are being neglected. In order to mathematically describe this process, we take a partial trace of the complete system, and atom(s) still interacting with the resonator, labeled ρs, tracing out the departed atoms’ degrees of freedom,

ρs = Tra ρtot , (6.19)

109 4.0 3 atoms 1 atom 2 atoms 2 atoms enter enters enter enter 3.5

3.0

2.5

2.0 Photon Number Atom 1 State 1.5 Atom 2 State 1.0 Atom 3 State Number of excitations 0.5

0.0

0 250 500 750 1000 1250 1500 1750 2000 Time t (arb. unit)

Figure 6.4: Schematic diagram of the average photon occupation number of the microwave resonator with atoms being deterministically monitored for a single realization. The moments atoms enter the resonator are marked by grey, dashed gridlines. Three atoms enter at a time t = 0, the state of one of them being displayed by the dashed blue curve. Two atoms leave and one enters at t = 500. At a time t = 1000, the atom 1 leaves the resonator, and a diﬀerent atom 2 enters, whose dynamics are displayed by green dashed curves, and two additional atoms are introduced. At t = 1500 the same atom remains and its state is displayed, while 2 new atoms enter, one of which is being displayed by the red dashed curve.

of the full density matrix ρtot. Thus, any new N atoms arriving at the resonator lead to a new initial density matrix

ρ = ρs ψa ψa ψb ψb ψN ψN , (6.20) 0 ⊗ | ih | ⊗ | ih | ⊗ · · · ⊗ | ih |

110 where the atoms arriving are in the ground state ψa = g . Fig. 6.4 shows | i | i that at any time during the coherent evolution, which follows a Rabi cycle, the resonator’s average number of photons has decreased. Once the atoms leave the microwave resonator after an average interaction time τ¯, which can be estimated from the atom beam’s velocity distribution and is experimentally known, the resonator ends up in a thermal state with a lower average number of photons than when the atoms began interacting with the resonator. Therefore, by continuously reintroducing ground state atoms to the system, the resonator mode can be effectively cooled. Since the Rabi frequency for a larger number of atoms has a larger amplitude, this approach leads to a more effective cooling process than when at most one atom interacting with the resonator is considered. At the point where the a new atom is introduced to the microwave resonator, an additional photon is absorbed by the atom within its Rabi cycle, leading to a change in the state of the microwave resonator and lowering the average number of microwave photons. Therefore, a successive introduction of atoms into the resonator leads to an effective cooling of the resonator mode to which they are coupled.

6.2.4 Ensemble average of the trajectories

To quantify the cooling of the microwave resonator mode, the relevant experimental parameters are the average interaction time, τ¯ = 50 ns, the interaction strength, g/(2π) = 4.25 MHz, as well as the dissipation rate of the microwave resonator, γ/(2π) = 3 kHz, which is derived from the res-

111 onator’s Q-factor of Q = 106 for a fundamental frequency of ν = 18.78 GHz. In this process we take into account many realizations of the possible trajectories of a single experimental run. All possible realizations are then averaged at each time step, thus giving a quantiﬁable time-resolved expectation value that would be observable when running the experiment. The average number of thermal microwave resonator photons, or phonons in a mechanical oscillator, can be expressed as [75]

1 n th = ~ω . (6.21) h i e kBT 1 − In the case of a resonance frequency of ν = 18.78 GHz, the average number of photons in the environment at a temperature of T = 4 K is n 3.96. h ith ≈ As can be seen from Fig. 6.5, in the case of a sparse beam of atoms, i.e., r = 1 MHz, the numerical calculations reproduce accurately the behavior obtained with the analytical method. Increasing the rate of atoms crossing the resonator to 15 MHz corresponds to a regime where the analytical method breaks down. The effective cooling of the fundamental resonator mode with a dense beam of Rydberg atoms prepared in the lower of two circular states is illustrated in Fig. 6.6. In this case the probability of more than one atom interacting with the resonator at a time is not negligible anymore but the cooling mechanism is more effective. This is due to the fact that the more atoms cross the resonator per second, more excitations can be carried away. Therefore, a steady state can be reached where the average number of photons has been significantly reduced to as low as n ss = 0.006 within a time of t 20 25 µs. The parameters considered in h i ≈ − this study are expected to be accessible with current experimental hybrid architectures [80]. In Fig. 6.7 the effect of individual atomic trajectories on

112 4.0

n 3.5

e 3.0 b m

u 2.5 N

o 2.0 t o h

P 1.5

e g

a 1.0 r e

v Numerical Method

A 0.5 Analytical Method 0.0 0.01 0.1 1 10 100 Time t ( s)

Figure 6.5: Comparison of the analytical and numerical method of treating the coupled atom-resonator system in a parameter regime in which both approaches are valid. The rate of Rydberg atoms crossing the resonator is

6 1 r = 10 s− , the constant interaction time is τ = 50 ns, and the interaction strength is g/(2π) = 4.25 MHz. the resonator photon occupation number are plotted and compared to the ensemble average of 100 trajectories. The individual trajectories show the quantum jumps in the average photon number, illustrating the departure of an excited atom, which effectively lowers the average photon number in the resonator. From the results provided here we conclude that significant gains in resonator cooling efficiency can be achieved by increasing the flux of atoms interacting with the resonator such that more than one interacts at a time. For resonators with dimensions of 10 50 µm, and typical atom beam den- −

113 4.0

n 3.5

e 3.0 b m

u 2.5 N

o 2.0 t o

h 1.5 P

g 1.0 a r e

v 0.5 A

0.0

0.01 0.1 1 10 Time t ( s)

Figure 6.6: The eﬀect of a dense beam of atoms prepared in the lower of two circular Rydberg states on the thermal photon occupation number in the resonator. The rate of Rydberg atoms crossing the resonator is r = 15 MHz, the constant interaction time is τ = 50 ns, and the interaction strength is g/(2π) = 4.25 MHz.

8 3 sities of 10 cm− , as were measured in Chapter 3, this regime in which a ∼ few (2 5) atoms interact with the resonator at any given time is achievable. − A similar method of using Rydberg atoms to de-excite coplanar waveguide resonators has recently been published by Sarkany et al. [99], where an ensemble of atoms in the ground state is trapped above the resonator, which extracts photons by continuous optical pumping. The eﬃciency of both methods is compared in Fig. 6.8 for the same experimental parameters. The cavity mode resononance frequency is ω/(2π) = 5 GHz and its quality factor is assumed to be Q = 105, resulting in a decay rate of

114 (a) First calculated trajectory. (b) Second calculated trajectory.

4.0 4.0 1st Trajectory 2nd Trajectory n n 3.5 100 Trajectories 3.5 100 Trajectories

r r e e b b m m 3.0 3.0 u u N N

n n o 2.5 o 2.5 t t o o h h P P

e 2.0 e 2.0 g g a a r r e e v 1.5 v 1.5 A A

1.0 1.0 0 0.5 1 1.5 2 0 0.5 1 1.5 2 Time t ( s) Time t ( s)

Figure 6.7: Two calculated trajectories for individual atoms compared to the average of 100 simulated trajectories, displaying the quantum jumps in the photon ﬁeld induced by an atom entering the resonator and being excited.

(a) Method by Sarkany et al. [99] (b) Numerical atom beam method.

1.0 1.0

0.8 0.8

0.6 0.6

0.4 0.4 Probability p ( n ) Probability p ( n )

0.2 0.2

0.0 0.0 0 1 2 3 4 5 0 1 2 3 4 5 Photon Number n Photon Number n

Figure 6.8: Comparison of the photon probability distribution of the method a) studied by Sarkany et al. [99] and b) the method investigated in this chapter with an atomic rate of r = 145 106 s 1 and an average · − interaction time of τ¯ = 10 ns. Both concepts lead to a sharp peak around n = 0 and a signiﬁcantly lower average eﬀective photon number in the steady state.

115 Trajectories:500 5

Average Photon Number 1

0 100 101 102 103 Time in ns

Figure 6.9: Time-dependence of the average resonator photon number.

An average steady state photon number of n ss & 0.1 can be reached h i within a time of t 1 µs, utilizing approximately 150 Rydberg atoms to ≈ eﬀectively cool the resonator.

γ/(2π) 50 kHz. The coupling strength between resonator and atom is ≈ g/(2π) = 5 MHz. Considering these parameters, and assuming an atomic

6 1 beam rate of r = 145 10 s− with an average interaction time of τ¯ = 10 ns, · the average number of photons achievable in the resonator is n ss & 0.1, h i compared to n ss = 0.5 given in [99]. The time-resolved evolution of the h i average photon number in the resonator is displayed in Fig. 6.9, illustrating the rapid eﬀective cooling. Based on this timescale, the average number of Rydberg atoms crossing the resonator to achieve this cooling eﬀect is estimated to be approximately 150 atoms.

116 Chapter 7

A Microwave Resonator-Nanomechanical Oscillator–System driven by an Atomic Beam

Recently, one particular focus of hybrid quantum system research has been the proposal of quantum state preparation of an individual system with the help of another quantum system. The methods described and ana- lyzed in Chapters 5 and 6 to cool selected modes of coplanar microwave resonators are in principle quite general. Because of the wide range of Rydberg-Rydberg transition frequencies available in any given atom or molecule they can be implemented over a wide range of resonance frequencies. With this in mind the formalisms developed in Chapter 5 and Chapter 6 are extended in this chapter to cool selected modes of nanome-

117 chanical resonators. The resonators considered have resonance frequencies in the microwave regime of the electromagnetic spectrum. The types of nanomechanical resonator devices of interest are therefore those based on micrometer membrane [107] or toroidal microresonator technologies [112].

7.1 Derivation of the Hamiltonian

To treat the coupled Rydberg-atom–optomechnaical-resonator system, the starting point considered is represented by the Hamiltonian given and simpliﬁed in [19],

H = H0 + Hac + Hmc + HL, (7.1) where H0 are the system Hamiltonians of the atoms, photon and phonon modes involved, Hac is the Hamiltonian describing the coupling between atoms and cavity, Hmc represents phonon-photon interaction of the nanomechanical resonator and the cavity, and HL is the laser drive of the pump mode of the cavity. The Hamiltonian (7.1) for the combined Rydberg-atom–optomechanical- oscillator can be expressed in detail as

ωa X (i) H = ω b†b + ω a†a + σ + ω a†a , (7.2) 0 m 0 2 z p p p i X (i) (i) Hac = gi aσ+ + a†σ , (7.3) − i

H = gm(b + b†)(a†a + a†ap a†ap a†a), (7.4) mc p − − p

iωLt iωLt HL = Ωpe− ap† + Ωp∗ e ap. (7.5)

Here, a two mode cavity system with frequencies ω0 and ωp mediates an excitation transfer between a moving membrane, toroidal microresonator

118 or patterned nanobeam (i.e. nanomechanical oscillator) and the atom(s).

In order to enhance the radiation pressure coupling g0 between the mechanical oscillator and the resonator, a laser with frequency ωL and Rabi- frequency Ωp continuously pumps the cavity. This laser-pump frequency

ωL is strongly detuned from the atomic transition frequency ωa, which is close to or on resonance, respectively, with the second cavity mode ω0. However, we are ignoring the dipole-dipole interaction between the atoms itself. Transforming these expressions into a frame of reference rotating with respect to the laser drive by applying the unitary transformation (i) P σz U = exp iωLt a†a + a†ap + , the time-dependence in the driving − p i 2 term is removed via the application of a lemma following from the BCH formula,

sX sX e Ye− = Y + s[X , Y], (7.6) if [X , [X , Y]] = [Y , [X , Y]] = 0. We consider the regime of a strong laser pump of the auxiliary cavity mode, Ωp gm, i.e., the coupling of photons of this auxiliary cavity mode to mechanical resonator phonons is weak. Here the pump mode of the cavity is close to a coherent steady state α apart from small quantum

2 fluctuations δa†δap α , stemming from nonlinearities introduced by h p i | | the nanomechanical resonator [48]. This allows for the replacement of the operator ap α + δap, and leads to a linearization of the radiation pressure → coupling in Hmc. The modified equilibrium position of the nanomechanical oscillator can be expressed as b β+b. First the zero point energy is shifted, → which can be done because a global phase shift does not affect the dynamics of the system. Applying the displacement operator D(α) = exp (αa† α∗a) −

119 to the Hamiltonian leads to a unitary transformation of the form

D(α)aD(α) = a + α. (7.7)

Therefore choosing appropriate values for the displacements r ω α = m β, (7.8) − gm

Ωp + ∆p (β + β∗) = , (7.9) − gm and including only enhanced optomechanical interaction strengths of or-

2 ( ) ( ) der gα and gα eliminates the ﬁrst oder terms in b † and δa † . Hence, we | | | | p obtain H = ∆ + gm(β + β∗) a†a Ωpδa†δap + ωmb†b 0 − p ∆a X (i) + σ + H + g (b + b†)(α∗δa + αδa†) 2 z al m p p i

gm(b + b†)(α∗a + αa†) gm(β + β∗)(α∗a + αa†). (7.10) − − Further transforming the Hamiltonian into a frame of reference rotating with respect to (Ωpδa†δap + ωmb†b) and assuming gmα Ωp ωm , which p | | | − | is self-consistenlty fulﬁlled due to the strong pump laser drive and a small single photon-phonon interaction strength, leads to fast oscillating terms

( ) in the coupling strength associated with the b † δap interactions. These fast oscillating terms can be neglected under the rotating wave approximation (RWA) [4], and doing this leads to the result that

∆a X (i) H ∆˜ a†a + ∆˜ pδa†δap + ωmb†b + σ + Hal ≈ 0 p 2 z i

gm(b + b†)(α∗a + αa†) gm(β + β∗)(α∗a + αa†). (7.11) − −

The δap†δap-terms are completely decoupled from the rest of the system, and are therefore ignored in the following by performing a partial trace of

120 the density matrix with respect to the degrees of freedom of the photons with frequency of the driving field. Thus, the influence on the dynamics of the entire system will simply be a global phase-shift. Applying another displacement operator unitary transformation, U = exp(θa θa†), elimi- − ( ) gm(β+β∗)α nates the linear terms in a † for an appropriate choice of θ = . It also ∆˜ 0 does not affect Hal because prior to applying the RWA it can be written as

Hal = igac(σ+ +σ )(a a†), which commutes with the unitary transformation. − − ( ) The resulting linear terms in the b † -operators can be absorbed into their displacement again. This is accomplished analogously to the procedure above by adjusting the term for the pump mode laser intensity in Eq. (7.8),

ω α 2 2θ m β. (7.12) | | → − gm

By assuming αgm ∆˜ + ωm , whereas ∆˜ ωm is of the same order of | | | 0 | | 0 − | magnitude as αgm , we can apply a further RWA where we can neglect the | | rapidly oscillating counter-rotating terms ba and b†a†, and then transform back, to obtain

X X ˜ ∆a (i) (i) (i) H ∆0a†a + ωmb†b + σz + G(a†b + ab†) + gi aσ+ + a†σ , (7.13) ≈ 2 − i i where the photon-enhanced radiation pressure coupling strength is rede- p ﬁned as G = gmα. Because α Np, and assumed to be real-valued for − ∝ convenience but without loss of generality, this means that the interaction strength between the nanomechanical oscillator and the cavity is proportional to the intensity of the pump laser. In the calculations and ﬁgures in the Chapters 7.2 and 7.3 we assume the following parameters, unless otherwise stated, in terms of the nanomechanical oscillator frequency: a frequency of the microwave resonator of

121 ωc/ωm = 2, a single-photon optomechanical coupling strength of gm/ωm =

4 5 2.5 10− , that together with a laser-drive generating nd = 1.6 10 pho- · · tons enables a light-enhanced coupling strength of G/ωm = 0.1, a Rydberg atom transition frequency of ωa/ωm 2, and atom-resonator coupling ≈ strength g/ωm = 0.1. Achievable experimental parameters for the individual quantum systems fall into the same orders of magnitude, as can be seen in Table 1.2, when a patterned Si nanobeam [25] is considered as the nanomechanical oscillator. However, a hybrid architecture of a combination of these three individual quantum systems with the same parameters as considered here has not yet been developed. The study in this Chapter therefore serves as a proof of principle. In the following we aim to diagonalize the photon-phonon–subsystem by moving to a joint polariton-picture, which allows for the theoretical framework developed in Chapter 5 and Chapter 6 to be applied to the coupling of this subsystem to an atomic beam.

7.2 Microwave Resonator-Mechanical Oscillator–

Subsystem

To study the photon-phonon–subsystem comprised of an optical cavity and a nanomechanical resonator, the Hamiltonian H = Hmc + Hac where

Hmc = ∆˜ 0a†a + ωmb†b + G(a†b + ab†), (7.14) can be exactly diagonalized within a new basis set of polariton excitations because the total particle number N = a†a + b†b is conserved for this

122 subsystem:

[Hcm , a†a + b†b] = 0 (7.15)

( ) ( ) ( ) ( ) where [a † , a†a] = ( )a † and [b † , b†a] = ( )b † . This means that the eigen- − − values of the Hamiltonian Hcm can be obtained in terms of Bogoliubov modes via the canonical transformation                   a  cos φ sin φ α         =     . (7.16)             b  sin φ cos φ β − The diagonalized Hamiltonian expressed in terms of the new modes is therefore                    ˜          ∆0 G  a ωα 0  α       !             =       , (7.17) a† b†     α† β†              G ωm b  0 ωβ β where we need to solve the equation system following from                    ˜      cos φ sin φ ∆0 G   cos φ sin φ ωα 0   −              =   . (7.18)                 sin φ cos φ   G ωm  sin φ cos φ  0 ωβ − This equation system has four diﬀerent sets of solutions. For convinience the set for which both sin φ, cos φ > 0 is chosen for the canonical transformation. We deﬁne the detuning ∆ = ωm ∆˜ , Rabi frequency Ω = − 0 2 2 √∆ + 4G and average polariton energy E0 = (∆˜ 0 + ωm)/2 to simplify the

123 (1)

m 1 / E

y g r 0 e n e n e g i

E -1

-1 -0.5 0 0.5 1 Detuning / m

(2a) 1

Photon Excitation 0.5 Phonon Excitation Relative Amplitude

0 -1 -0.5 0 0.5 1 Detuning / m

(2b) 1

Photon Excitation 0.5 Phonon Excitation Relative Amplitude

0 -1 -0.5 0 0.5 1 Detuning / m

Figure 7.1: (1) The dependence of the eigenenergies of both polariton- modes is displayed when the detuning between the microwave resonator and the laser drive ∆˜ 0 is decreased from 2 ωm to 0 ωm by varying the laser drive frequency. (2a) and (2b) show the relative amplitudes of the photonic and phononic excitation.

124 expressions to r Ω + ∆ cos φ = , (7.19) 2Ω r Ω ∆ sin φ = − , (7.20) 2Ω Ω ω = E , (7.21) α 0 − 2 Ω ω = E + , (7.22) β 0 2 where the polaritons are a combination of photons and phonons.

7.3 Microwave Resonator-Mechanical Oscillator-

Atom–System

Rewriting the full Hamiltonian in Eq. (7.13) in terms of the Bogoliubov- polariton modes, and considering for the sake of simplicity the special case of only one atom interacting with the resonator,

∆a H = ωαα†α+ωββ†β+ σz+g cos φ ασ+ + α†σ +g sin φ βσ+ + β†σ (7.23) 2 − −

It can be seen in Fig. 7.1 that for each atomic (de-)excitation one polariton- mode is (created) annihilated, but with a diﬀerent composition of photon- and phonon-parts. A completely symmetric state would be reached for ∆ = 0, because in that case sin φ = cos φ. The commutator between the total excitation number (polariton number and atomic excitation) and the Hamiltonian is

σz [α†α + β†β + , H] = 0. (7.24) 2

125 Because the particle number is conserved, the full Hamiltonian can be diagonalized within this 3-particle-excitation subspace. One point to note in doing this is the fact that this new quasiparticle, a three-particle-polariton, is a mixture of two bosonic polaritons and a fermion. Therefore it is ﬁrst necessary to transform the problem into an interaction picture according to

Hˆ i = Uˆ r f (Hˆ Hˆ )Uˆ † , (7.25) − 0 r f with a rotating frame at the detuning of the atomic frequency,

Uˆ r f = exp [i∆atHˆ 0] (7.26) σˆ z = exp i∆ t αˆ †αˆ + βˆ†βˆ + . a 2 Following this, the interaction Hamiltonian can be written in matrix form as              ∆α 0 g cos φ  α                  Hˆ i =       , (7.27) α† β† σ+  0 ∆β g sin φ  β                  g cos φ g sin φ 0  σ  − where the detunings ∆ = ω ∆a represent the frequency diﬀerence be- α,β α,β − ( ) ( ) tween the polariton modes α † , β † of the microwave resonator-mechanical oscillator subsystem and the detuning of the atomic transition frequency from the frequency of the pump laser. The eigenvalues of this Hamiltonian can be readily calculated and exhibit an avoided crossing for the parameters fulﬁlling all approximations, as seen in Fig. (7.2). By changing to a basis corresponding to the 3-particle-excitation modes xˆ, yˆ, zˆ, which is done

126 1

0.5 m / E

y g r 0 e n e n e g i

E -0.5

-1 -1 -0.5 0 0.5 1 Detuning / m

Figure 7.2: The ﬁrst 3 eigenenergies of the Hamiltonian Hi depending on the detuning between the microwave resonator detuning and the me-

chanical oscillator ∆ = ∆˜ 0 ωm. The detuning between the microwave − resonator and the laser drive ∆˜ 0 is decreased from 2 ωm to 0 ωm. via the canonical transformation              αˆ  xˆ              ˆ  = Rˆ   (7.28)  β  yˆ                 σˆ  zˆ, −

127 the Hamiltonian can be rewritten. As a result, the eigenvalues of the polariton-modes are numerically calculated from             ωx 0 0   ∆α 0 g cos φ               = Tˆ †   Tˆ. (7.29)  0 ωy 0   0 ∆β g sin φ                  0 0 ωz g cos φ g sin φ 0 

The results for ω(x,y,z) give the eigenenergies of the Hamiltonian and therefore its resonance modes. The transformation matrix Tˆ, consisting of the eigenvectors of the Hamiltonian, gives information about their polaritonic composition, revealing if these energies are mostly dominated by atomic, phononic or photonic excitations. The rotation matrix mixing the three particle types is given by the eigenvectors of the Hamiltonian Hi that have been previously plotted in Fig. 7.2. The composition of the quasiparticles representing the eigenenergies is displayed in Fig. 7.3. It can be seen that the ﬁrst two three-particle-polaritons, graphs (a) and (b), are an antisymmetric mixture of atomic excitation and either α-polariton, the blue curve (a), or β-polariton, which is the red curve (b). The third quasiparticle, displayed by the green curve (c), shows a transition from being dominantly a β-polariton, to mostly atomic excitation at zero detuning between the polaritons, ∆ = 0, This information is crucial as it dictates the parameter regime in which the polariton modes are mostly phononic. In order to engineer a cooling mechanism for the phononic mode of the mechanical oscillator via a beam of atoms, in the next section we exploit the interaction between the static microwave resonator-mechanical oscillator-subsystem and the atoms prepared in the lower circular Rydberg state to de-excite

128 (a) 1

0.8

0.6 -polariton -polariton 0.4 atomic excitation

Relative Amplitude 0.2

0 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 Detuning / m

(b) 1

0.8

0.6 -polariton -polariton 0.4 atomic excitation

Relative Amplitude 0.2

0 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 Detuning / m

0.4

Relative Amplitude 0.2

0 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 Detuning / m

Figure 7.3: The quasiparticle-composition of the eigenenergies (a) ωx, (b)

ωy and (c) ωz in Equ. (7.29) of the interaction Hamiltonian, Hi, that are plotted in Fig. 7.2.

129 such a mainly phononic polariton mode.

7.4 The Atomic Beam as an engineered environ-

ment

The Hamiltonian of the complete atom-polariton system can be split into two parts, one with and one without the atomic interaction:

H = H0 + Hi, (7.30)

ωa H = ω α†α + ω β†β + σ , (7.31) 0 α β 2 z Hi = g cos φ ασ+ + α†σ + g sin φ βσ+ + β†σ . (7.32) − −

A necessary prerequisite to apply the theoretical framework established in Chapter 5 to mathematically describe a switchable interaction is that the commutator between the system Hamiltonian and the interaction Hamil- tonian vanishes. This commuter is calulcated to be

[H0, Hi] = g cos φ(ωα ωa)(α†σ ασ+)+ g sin φ(ωβ ωa)(β†σ βσ+). (7.33) − − − − − − If it is assumed that the atomic transition frequency is resonanct with one of the two polariton modes, then

Either ω ωa = 0 • α − r Ω2 Ω∆ [H0, Hi] = g − (β†σ βσ+). (7.34) 2 − − gG If ∆ 0, it follows that [H0, Hi] (β†σ βσ+). Therefore, an upper ≈ ≈ √2 − − limit can be given in the regime ∆ 0 for the commutator ≥ gG [H0, Hi] (β†σ βσ+). (7.35) ≤ √2 − −

130 Alternatively, for ω ωa = 0 and ∆ 0 a limit is analogously obtained • β− ≤ to be gG [H0, Hi] (ασ+ α†σ ). (7.36) ≤ √2 − − From this it can be seen that the sign of the detuning determines which polariton mode is close to resonance with the atomic transition frequency. Following this, an upper/lower limit can be obtained for the commutator between the full Hamiltonian and the atomic Hamiltonian. In order to determine which regime to consider to most effectively cool the phonon- mode, the combination of the three polariton types in Fig.(7.3) has to be investigated. This can be done by plotting both regimes considered, the eigenenergies of the interaction Hamiltonian within these regimes, and the composition of the excitations of the polariton types following from the eigenvectors of this calculation. These eigenvectors then need to be additionally rotated around the atomic-excitation axis with a 3D rotation matrix, according to their polariton-rotation when diagonalizing the mechanical oscillator-resonator subsystem. The result is the mix of photon-, phonon- and atom-excitations of the eigenstates of this interaction Hamil- tonian. Whichever eigenenergy contains the highest proportion of phonon- and atom-excitations then has to be cooled by removing a quasiparticle, which is achieved by removing the atomic excitation of the polariton via the beam of atoms. This is going to establish the regime of experimental parameters leading to the most effective cooling procedure of the mechanical oscillator mode. Assuming a sufficiently small coarse grained time scale, ∆t, so that q gGn ∆t 1, (7.37) (α,β) 131 the commutator in Eq. (7.33) is negligible and the time evolution operator can be approximated as

i∆tH U = e−

2 i∆tH0 i∆tHa O (∆t √gG) = e− e− e (7.38) ···

i∆tH i∆tHa e− 0 e− . ≈ Thus, the formalism introduced in Chapter 5 can be applied. On a suﬃ- ciently small coarse grained time-scale the time-evolution of the density- matrix of the whole system without an atom present can be written as

iHˆ 0(∆t) iHˆ 0(∆t) ρˆ0(t + ∆t) = e− ρˆ(t)e , (7.39) and with an atom present as h i ( iHˆ 0∆t) ( iHˆ aτ) (iHˆ aτ) (iHˆ 0∆t) ρˆ1(t + ∆t) = e − e − ρˆ(t)e e , (7.40) where τ is the interaction time of a single atom with the mechanical oscillator-resonator–system. By combining these two equations, considering the rate of ground state atoms r and excited state atoms R crossing the resonator per second and transforming to an interaction picture with respect to H0, we arrive at an equation determining the density matrix at time t + ∆t given the density matrix at time t, which can be expressed as Z i ˆ aτ/~ i ˆ aτ/~ ρˆ (t + ∆t) = ρˆ (t) (1 r∆t R∆t) + r∆t dτPg(τ) e− H ρˆ (t)e H tot tot · − − tot Z i ˆ aτ/~ i ˆ aτ/~ +R∆t dτPe(τ) e− H ρˆtot(t)e H , (7.41) where τ is the interaction time of a single atom with the microwave resonator-mechanical oscillator system and Pg,e(τ) is the distribution of interaction times for the ground and excited state, respectively. Before an

132 atom prepared in the lower circular Rydberg state interacts with the system the total density matrix can be factorized into a density matrix describing the state of the system, ρˆ and a density matrix g g describing the pure cm | ih | state of the atom at arrival, as ρˆ = ρˆ g g . In the case of an atom tot cm ⊗ | ih | prepared in the excited circular state, the full density matrix factorizes as

ρˆtot = ρˆcm e e before the interaction. Hence, it is necessary to investigate ⊗ | ih | the eﬀect of the atomic interaction Hamiltonian on the ground and excited states, respectively, of a two level system (i.e. atom in a circular Rydberg state). In the case of atoms prepared in the lower circular state,

X ( iτg)kHˆ k ( igHˆ aτ) a e − g = − g , (7.42) | i k! | i k with Ha = cos φα + sin φβ σ+ + cos φα† + sin φβ† σ , (7.43) − 2 2 2 H = cos φ sin φ(α†β + αβ†) + cos φα†α + sin φβ†β + e e , (7.44) a | ih | where the substitution Hˆ a gHˆ a has been made and the interaction → strength between atom and photon-ﬁeld g has been taken out of the atomic interaction Hamiltonian. The exponential series in Eq. (7.42) is extended up to second order in gτ¯ 1, where τ¯ is the average interaction time of an atom with the system. This approximation is fulﬁlled in the weak- coupling regime where the perturbation of the system due to the interaction with the atom is small. Evaluating the exponentials to second order and taking the partial trace with respect to the atomic degrees of freedom, ρˆ (t) = Tr ρˆ (t) , we arrive at an equation giving the change of the re- cm a{ tot } duced density matrix of the system due to an interaction with a beam of atoms (with at most one atom interacting with the system during each time

133 step ∆t). Averaging over the distribution, assuming here for simplicity that we can replace these terms by the average interaction time τ τ¯, we can → rewrite the equation of motion for the reduced density matrix so that it closely resembles the form of the Lindblad quantum master equation, i.e.,

2 2 2 ∂ρˆcm(t) rg τ cos φ = + 2αρˆcm(t)α† ρˆcm(t)α†α α†αρˆcm(t) ∂t atom 2 − − rg2τ2 sin2 φ + 2βρˆcm(t)β† ρˆcm(t)β†β β†βρˆcm(t) 2 − − rg2τ2 sin φ cos φ + 2αρˆcm(t)β† ρˆcm(t)αβ† αβ†ρˆcm(t) 2 − − rg2τ2 sin φ cos φ + 2βρˆcm(t)α† ρˆcm(t)α†β α†βρˆcm(t) 2 − − Rg2τ2 cos2 φ + 2α†ρˆcm(t)α ρˆcm(t)αα† αα†ρˆcm(t) 2 − − Rg2τ2 sin2 φ + 2β†ρˆcm(t)β ρˆcm(t)ββ† ββ†ρˆcm(t) 2 − − Rg2τ2 sin φ cos φ + 2α†ρˆcm(t)β ρˆcm(t)α†β α†βρˆcm(t) 2 − − Rg2τ2 sin φ cos φ + 2β†ρˆcm(t)α ρˆcm(t)αβ† αβ†ρˆcm(t) . 2 − − (7.45)

In the context of open quantum systems, the eﬀect of the atom on the system is treated here as an interaction with the environment. The result is a quantum master equation in Lindblad form resulting from the interaction with a continuous beam of atoms. This can be linearly combined with the coherent evolution of the closed system, H0, given by the van Neumann equation, where the partial trace has been taken over the atomic degrees of freedom. In doing this the ωz-part turns into a constant value which can be absorbed by a shift of the zero-point energy. This leads to a master

134 equation of the form

d ∂ρˆcm(t) ρˆcm(t) = i H0, ρˆcm(t) + . (7.46) dt − ∂t atom

An additional possibility is considering other environments for the microwave resonator or mechanical oscillator, such as for example interacting with the thermal state of the environment [21].

7.4.1 Steady State of the system interacting with an atomic beam

As a ﬁrst step in applying this formalism to the microwave resonator- mechanical oscillator system interacting with an atomic beam, we investigate the average polariton numbers that can be calculated. Naturally, with the master equation derived above, and assuming that the mechanical oscillator and microwave resonator are otherwise closed systems, we

( ) ( ) calculate the time-evolution of average values of interest, αˆ † , βˆ † and h i h i especially ( ) d d d n o d nˆ = αˆ †αˆ = Tr α†αˆρˆcm(t) = Tr α†αˆ ρˆcm(t) . (7.47) dth αi dth i dt dt

This can be done in a straight-forward manner by evaluating the expression with the time-derivative of the reduced density matrix, exploiting the cyclic property of the trace operator.

135 The set of equations of motion are therefore

d 2 2 2 αˆ †αˆ = g τ cos φ(R r) α†α dth i − h i g2τ2 sin φ cos φ + (R r) α†β + αβ† 2 − h i h i +Rg2τ2 cos2 φ, (7.48)

d 2 2 2 βˆ†βˆ = g τ sin φ(R r) β†β dth i − h i g2τ2 sin φ cos φ + (R r) α†β + αβ† 2 − h i h i +Rg2τ2 sin2 φ, (7.49)

d (R r)g2τ2 αˆ †βˆ = − + i(∆ ∆ ) α†βˆ dth i 2 α − β h i (R r)g2τ2 sin φ cos φ + − α†α + β†β 2 h i h i +Rg2τ2 sin φ cos φ, (7.50)

d (R r)g2τ2 αˆβˆ† = − i(∆ ∆ ) αβˆ† dth i 2 − α − β h i (R r)g2τ2 sin φ cos φ + − α†α + β†β 2 h i h i +Rg2τ2 sin φ cos φ. (7.51)

Here, the two competing processes of atoms prepared in the higher circular Rydberg state introducing excitations into the system and atoms prepared in the lower circular state extracting excitations lead to a detailed balance for a non-equilibrium steady state number of polaritons. We calculate the ! ! steady state of the system, d O = 0, because of d ρˆ = 0. The steady state dt h i dt is reached after a time t . For the microwave resonator-mechanical → ∞ 136 Cooling Polariton Excitations 102 5.0

2.5 101

0.0 1 0.5 0 100

10 1 Number of Polariton Excitations

10 2

1 0.8 0.6 0.4 0.2 0 Atom Beam Rate R/r

Figure 7.4: The eﬀective cooling process for the polariton excitations

α α ss + β β ss. Note that the x-axis is inversed in order to illustrate the h † i h † i eﬀect for an increase in the rate of atoms prepared in the lower circular Rydberg state. The inset shows the same calculation with a linear y-axis. oscillator system coupled to a reservoir of atoms via an atomic beam, we obtain for ∆α , ∆β, R α†α ss = β†β ss = . (7.52) h i h i r R − In this case, the steady state does not contain any buildup of correlations between the diﬀerent types of polariton because α†β ss = 0. The number of h i polaritons in the steady state thermalizes with respect to the ratio of ground state and excited state atoms interacting with the microwave resonator- mechanical oscillator subsystem. From this one can immediately see that for an atomic beam where R = 0 atoms are prepared in the excited state, a

137 steady state of the system is eventually reached which contains neither α- nor β-polaritons. This therefore represents a possible way to eﬀectively cool the nanomechanical oscillator. This result is not surprising considering we did not include the possibility of thermal heating due to the ﬁnite temperature of the environment.

7.4.2 Considering a Thermal Environment

To account for the coupling to a thermal environment it is assumed that the time of ﬂight of the Rydberg atoms is short compared to their coherence times. Therefore they arrive at the cavity in their initially prepared state. The cavity and mechanical resonator however experience thermal heating due to their environment. We must therefore investigate how their individual dissipation rates aﬀect the polariton states within the framework of the Lindblad master equation. The dissipator for the microwave resonator can be expressed as

γr γr D [ρ] = (N +1) 2aρa† a†aρ ρa†a + N 2a†ρa aa†ρ ρaa† , (7.53) r 2 r − − 2 r − − ( ) where the operators a † are the photon annihilation (creation) operators in the original Fock basis, γr is the dissipation rate of the resonator following from its quality factor Qr and Nr is the number of photons in the resonator at thermal equilibrium with the environment at a temperature T = 4 K. Analogously, the dissipator for the mechanical oscillator is

γm γm D [ρ] = (N + 1) 2bρb† b†bρ ρb†b + N 2b†ρb bb†ρ ρbb† . m 2 m − − 2 m − − (7.54) These dissipators are expressed in terms of the polariton operators α and

β. We can summarize these terms as the internal dissipator Di[ρ] = Dr[ρ] +

138 Dm[ρ] such that γm 2 γr 2 Di[ρ] = (N + 1) sin φ + (N + 1) cos φ 2αρα† α†αρ ρα†α 2 m 2 r − − γm 2 γr 2 + N sin φ + N cos φ 2α†ρα αα†ρ ραα† 2 m 2 r − − γm 2 γr 2 + (N + 1) cos φ + (N + 1) sin φ 2βρβ† β†βρ ρβ†β 2 m 2 r − − γm 2 γr 2 + N cos φ + N sin φ 2β†ρβ ββ†ρ ρββ† 2 m 2 r − − + sin φ cos φ γ (N + 1) γ (N + 1) αρβ† + βρα† r r − m m + sin φ cos φ γ N γ N α†ρβ + β†ρα (7.55) r r − m m 1 1 sin φ cos φ γ (N + ) γ (N + ) α†βρ + ρα†β + αβ†ρ + ραβ† . − r r 2 − m m 2 The average thermal number of microwave resonator-photons and mechanical oscillator-phonons in the system can be expressed as 1 n th = . (7.56) h i ~ωr,m e kBT 1 − Together with the dissipator resulting from the atomic interaction, rg2τ2 cos2 φ D [ρ] = 2αραˆ † ραˆ †α α†αρˆ at 2 − − Rg2τ2 cos2 φ + 2α†ραˆ ρααˆ † αα†ρˆ 2 − − rg2τ2 sin2 φ + 2βρβˆ † ρβˆ †β β†βρˆ 2 − − Rg2τ2 sin2 φ + 2β†ρβˆ ρββˆ † ββ†ρˆ 2 − − 2 2 +rg τ sin φ cos φ αρβˆ † + βραˆ † 2 2 +Rg τ sin φ cos φ α†ρβˆ + β†ραˆ

(r + R)g2τ2 sin φ cos φ α†βρˆ + ραˆ †β + αβ†ρˆ + ραβˆ † , − 2 (7.57)

139 which exhibits a symmetric structure. This leads to competing processes, the thermal heating of the system due to the environment and the cooling eﬀect of Rydberg atoms prepared in the circular ground state absorbing excitations from the system and carrying them away. Together with the fully diagonalized Hamiltonian for the system in the polariton basis, H0 = ωαα†α + ωββ†β, we can evaluate the equations of motion for the average polariton-numbers, which are described by the master equation

ρ˙ = i[ρ, H0] + Di[ρ] + Dat[ρ], (7.58) leading to the following set of equations of motion d 2 2 2 2 2 α†αˆ = γ N sin φ + γ N cos φ + Rg τ cos φ dth i m m r r 2 2 2 2 2 γ sin φ + γ cos φ + (r R)g τ cos φ α†α − m r − h i 2 2 γm γr (R r)g τ + sin φ cos φ + − α†β + αβˆ † , (7.59) 2 − 2 2 h i h i

d 2 2 2 2 2 β†βˆ = γ N cos φ + γ N sin φ + Rg τ sin φ dth i m m r r 2 2 2 2 2 γ cos φ + γ sin φ + (r R)g τ sin φ β†β − m r − h i 2 2 γm γr (R r)g τ + sin φ cos φ + − α†β + αβ† , (7.60) 2 − 2 2 h i h i

2 2 d (R r)g τ γr γm αˆ †βˆ = − − − + i(ω ω ) αˆ †β dth i 2 α − β h i 2 2 (R r)g τ γr + γm + sin φ cos φ − − α†αˆ + βˆ†β 2 h i h i + sin φ cos φ γ N γ N + Rg2τ2 (7.61) r r − m m ! d † = αˆβˆ† , (7.62) dth i

140 where it can be seen that the interaction between the two diﬀerent types of polaritons can be driven, and possibly controlled, by the rate of incoming atoms prepared in the higher circular Rydberg state, R, and lower circular Rydberg state, r. Evaluating this set of equations of motion in the steady state condition, ! d Aˆ = 0, assuming the microwave resonator has been prepared in its dt h iss vacuum state, N 0, as has been shown in Chapter 6, and assuming r ≈ the Rydberg atoms have been prepared in the lower circular state, R = 0, leads to the steady-state solutions. An analytical solution can be obtained. However, the solution is too convoluted to be given here and therefore only the dependence of the analytical steady state solution of the average polariton number on the ﬂux of atoms has been plotted in Fig. 7.5. The parameter set for the simulation is γm/ωm = 0.01, γr/ωm = 0.05, g/ωm = 10, G/ω = 15τ = 0.05ω , ∆˜ /ω = 5, and N = 40. A reduction of m − m 0 m − m the average polariton number can be observed, up to n 0.20 and h αiss ≈ n 0.45, respectively. The simulation carried out here serves as a h αiss ≈ proof of principle that optomechanical polariton modes can be emptied with an engineered reservoir of a beam of Rydberg atoms.

141 10 Phononic Polariton 8 Photonic Polariton

Polariton Number 〈 n 〉 2

0 0 1000 2000 3000 4000 5000

Atom Beam Rate r(inω m/1000)

Figure 7.5: The dependence of the two polariton branches on the rate of

the atomic beam,r, for Nm = 40. Because γr/γm = 5, the average polariton numbers of the initial, thermal state are α α 4.74 (mainly photonic h † ith ≈ mode) and β β 9.17 (mainly phononic mode). h † ith ≈

142 Chapter 8

Conclusions

In this thesis, ﬁrst the behavior of strongly polarized ensembles of Rydberg atoms has been studied. Electrostatic dipole-dipole interactions between Rydberg atoms were investigated to realistically model the energy shifts within an atomic ensemble. The results of the calculation were compared to experimental data, yielding good qualitative agreement. This combination of experiments and calculations allowed the demonstration of a macroscopic screening of electrical noise that arose as a result of the local polarization of the Rydberg atoms. Following this, the possibility to observe trapping states of a superconducting microwave resonator, and the implementation of quantum state preparation has been shown by varying controllable experimental parameters such as the ﬂux of atoms, their velocity and spatial distribution, or the detuning of the atomic transition frequency from the fundamental frequency of the resonator. These results have demonstrated, for example, that it is possible to measure the velocity distribution as well as the spatial distribution of a gas-phase ensemble of Rydberg atoms by monitoring the

143 state of the transmission line resonator interacting with the atomic beam. However, this formalism is limited to diffuse atomic beams with at most one atom interacting with the transmission line resonator at any time. A numerical method was then developed to show that in a realistic experimental setting the average number of thermal photons of the resonator, n 4, can be significantly reduced to n = 0.006 with a dense beam h ith ≈ h iss of atoms prepared in circular Rydberg states. This result paves the way for highly efficient single-photon experiments in hybrid quantum information processing. This numerical formalism, applied in order to simulate a dense beam of Rydberg atoms under realistic experimental conditions, allows for the measurement of correlations between the microwave resonator Fock states and a chosen number of atoms. This could lead to the realization of Schrödinger cat or squeezed states in the resonator once the optimal experimental parameters for the preparation of such quantum states have been identified. Finally, an analytical formalism has been derived to engineer a quantum reservoir of atoms coupled to an optomechanical system. Through this work, it has been shown that it is possible to empty optomechanical polariton modes with a beam of atoms. However, a more in-depth investigation needs to be carried out in order to realize the potential of this method. As a proposed next step in the research presented here, the application of the numerical formalism derived to describe the photon-phonon system could be extended to entangle specific states of the mechanical oscillator and microwave resonator, offering a route to study the implementation of protocols for quantum state transfer between photons and phonons me-

144 diated by Rydberg atoms. Additionally, a formalism could be derived for QND-measurements of the optomechanical system using Rydberg atoms. Another direction of interest is to include effects of electric dipole-dipole interactions between Rydberg atoms in the formalism developed to describe the coupling of a dense atomic beam to a microwave resonator. In general it is foreseeable to adapt and apply this numerical formalism to various hybrid quantum systems where the interaction with an atomic beam is continuously switched on and off due to the presence, or absence, of an atom. This atom does not have to be a pure two-level system, but could also be, e.g., a three-level lambda system. Extending the formalism in this way might open new possibilities for quantum state preparation, quantum computation and quantum communication with different hybrid quantum systems that are coupled to beams of atoms.

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163 Appendix A

Probability of N Atoms interacting with the Resonator

In this appendix a validation for the truncation of the numerical approach in Chapter 6 to at most ﬁve atoms interacting with the resonator at any time is provided. This is done by calulcating the probability of six or more atoms interacting with the resonator at the same time and showing that this probability is negligible within the context of the experimental parameters discussed.

As has been shown in Eq. (6.4), the probabilty of zero atoms entering the resonator can be calculated as 1 2r∆t p = − . (A.1) 0 1 r∆t − For any number of atoms n 1 arriving at the resonator within the same ≥ time step, ∆t, one can write

n pn = (r∆t) . (A.2)

164 Considering a time interval on the order of the interaction time, τ = k∆t, any permutation of n atoms that interact with the resonator during the interaction time must be calculated. Therefore, the probabilty of zero atoms interacting with the resonator over the time interval τ is

τ/∆t P0(τ) = (p0) (A.3) 1 2r∆tk = − . 1 r∆t − The probability of one atom interacting with the resonator during the time interval can be expressed as ! k k 1 P (k∆t) = (p ) − p 1 1 0 1 (A.4) k 1 1 2r∆t − = k − r∆t, 1 r∆t − where the binomial coeﬃcient prefactor accounts for the combinatorics, i.e., the k diﬀerent time steps in which the atom may enter the resonator. In order to calculate the probabilities of more than one atom interacting with the resonator at any given time, the binomial distribution must be extended to a multinomial distribution. This is due to the fact that, for example, in the case of n atoms, the probabilities of

all n atoms beginning to cross the resonator during the same time • step,

(n 1) atoms arriving during one time step and one atom arriving • − during a diﬀerent time step

(n 2) atoms arriving during one time step and either (i) two atoms • − arrive as a pair during a diﬀerent time step, or (ii) one atom each arrives at another diﬀerent time step

165 (n 3) atoms arriving during one time step and all combinations of • − the three other atoms arriving during other time steps

continuing via (n 4), (n 5), ... that must be extended to all the • − − atoms arriving each during a diﬀerent time step one by one must all be accounted for. For one realization of these cases, the probability of n atoms being multi- nomially distributed can be calculated as

(i + j + l + + z)! i j l z Pn(one realization) = ··· (p ) (p ) (p ) ... (pn) . (A.5) − i! j! l! ... z! 0 1 2

Here i is the number of time steps during which no atom enters the resonator, j is the number of time steps with one atom entering, m with two atoms entering and so forth, with

τ i + j + l + + z = (A.6) ··· ∆t summing up to the number of time steps the time interval, i.e., the interaction time, is split into. There exists a restriction for the possible values each number of time steps can take on. For example, the maximum number of time steps during which a single atom enters the resonator is j n. The ≤ maximum number of time steps during which two atoms enter is l n/2. ≤ For three atoms entering, the limit is m n/3. The limits for four and ≤ more atoms entering are calculated in similar fashion. The number of time steps during which no atom enters is then self-consistently obtained from Eq. (A.6) to be τ i = j + l + + z . (A.7) ∆t − ···

166 The probabilities for all possible, unordered combinations of i, j,... z N ∈ for a given number of atoms, n, are calculated in this manner and then added. The result obtained in this way is the probability, Pn(τ), of n atoms interacting with the resonator at the same time within the time interval of the interaction time, τ. Therefore the probabilities of having 2, 3, 4 or 5 atoms interact with the resonator at the same time over k time steps are expressed as

k(k 1) k 2 2 k 1 P (τ) = − (p ) − (p ) + k (p ) − p , (A.8) 2 2 0 1 0 2

k(k 1)(k 2) k 3 3 k 2 P (τ) = − − (p ) − (p ) + k(k 1) (p ) − p p 3 6 0 1 − 0 1 2 k 1 +k (p0) − p3, (A.9)

k(k 1)(k 2)(k 3) k 4 4 k(k 1)(k 2) k 3 2 P (τ) = − − − (p ) − (p ) + − − (p ) − (p ) p 4 24 0 1 2 0 1 2 k 2 k(k 1) k 2 2 k 1 +k(k 1) (p ) − p p + − (p ) − (p ) + k (p ) − p , (A.10) − 0 1 3 2 0 2 0 4

k(k 1)(k 2)(k 3)(k 4) k 5 5 P (τ) = − − − − (p ) − (p ) 5 120 0 1 k(k 1)(k 2)(k 3) k 4 3 k(k 1)(k 2) k 3 2 + − − − (p ) − (p ) p + − − (p ) − (p ) p 6 0 1 2 2 0 1 3 k(k 1)(k 2) k 3 2 k 2 + − − (p ) − p (p ) + k(k 1) (p ) − p p 2 0 1 2 − 0 2 3 k 2 k 1 +k(k 1) (p ) − p p + k (p ) − p . (A.11) − 0 1 4 0 5

Assuming coarse-grained timescale in the numerical simulation with a step size of ∆t = τ/100 and hence k = 100, and a value for the product of interaction time and atomic rate of rτ = 1.0, the probability of more than

167 1.4

1.2

(%) 1.0 n > 5 0.8

0.6

0.4 Probability P

0.2

0.0 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 Product r·τ

Figure A.1: The dependence of the probability of more than ﬁve atoms interacting with the resonator at the same time on the product of r τ. The · dashed lines indicate the maximum value of the product r τ for which · the error is 0.5%. ≤

ﬁve atoms interacting with the resonator at the same time is

Pn>5 = 1 Pn 5, (A.12) − ≤ where X5 Pn 5 = Pn(τ). (A.13) ≤ n=0 This leads to the result that

Pn 0.07% (A.14) >5 ≈ and validates the assumption to consider at most n = 5 atoms interacting with the resonator at any given time. It becomes clear that the validity of the truncation depends on the maximum atomic rate, r, and on the interaction

168 time, τ = k∆t. The dependence of the error of the approximation on the atomic rate, r, is plotted in Fig. A.1, and can be seen to be below . 0.5% for rτ . 1.45, which in theory is the limit for the atom rate of this speciﬁc numerical truncation. For a chosen interaction time of τ = 50 ns, the maximum atomic rate is calculated to be r = 29 MHz, and for an interaction time of τ = 10 ns, the maximum atomic would then be r = 145 MHz. Hence, the parameters for the calculations carried out in Chapter 6 are all within this limit.

169